Can the US borders, North and South, actually be sealed to prevent significant unauthorized crossing?

Sealed? NO
More difficult YES.
Then address the real issue.. They flock here because they are coddled and accepted once on the ground. Make Hiring them a felony and ENFORCE the law....at the Company LEADERSHIP level, not the manager or HR.
Deport them immediately.. like as in within 24 hours of a immigration court appearance that proves they are not here legally.
Capture, deport and require criminal action against the mules that bring in the illegals from the host country. Track the legal action, and a mule captured and returned that is not criminally charged in the home country .. we STOP immediately all foreign aid to that country.... Recall as debt owed all monies paid to that country. List their representatives in the US as persona non grata, deport, close the embassy if they have one until all the deported criminals are charged, and or funds returned. Any national in that host country that has assets in the US, seize those assists as partial Host country debt repayment notifying the former assets owner and country they the former assets owner should seek remediation and reimbursement from their country leadership.

The DoD has studied and planned against this problem for decades. It can be done. I was involved in a study in the 1980s. Our main concern was the movement of drugs, not people, across the southern border. We worked with the three Departments to determine the forces necessary to seal the Southern border. I don't remember the exact numbers, which may have been classified, but it takes a few Army Divisions, several Air Force squadrons, a fleet of Navy ships, and Coast Guard ships to do the job. It's not cheap. The rules of engagement get interesting because of restriction on using Active Duty military to enforce law. However, that problem can be overcome with an act of Congress--possibly harder to get than the funding for the force.

Technology that's come along since the 1980s will make enforcing the air and ocean approaches to the Southern coastline easier and more efficient. The number of Navy and Coast Guard ships and Air Force aircraft would be fewer now than in the 1980s. Let's face it, we know what a secure land border looks like. It looks like the Berlin Wall. We need the courage to admit we need the wall, the wire, the clear zones, the towers, and other physical security infrastructure. Add the technological enhancements including video cameras, motion sensors, listening devices, ground penetrating radar, and large and small drones and it decreases the number of soldiers or border guards necessary to prevent penetration compared to our 1980s estimate.

The Air Component Commander and Sea Component Commander need authorities to stop or shoot down/sink aircraft and watercraft believed to be moving to invade (important word) the United States. The Land Component Commander will need authority to use deadly force against people invading the US across the Mexican border. These are tough authorities to get and even more difficult to exercise, but I trust our senior military commanders to use appropriate force and good judgement to complete the mission.

Finally, we have to remain committed to the mission for decades. Every time we introduce a new defensive technology, tactic, or technique, those trying to penetrate the border (invade the US) will develop a courter. There's a lot of money in smuggling drugs and people. The criminals and terrorists can buy or steal the best technology and innovative planners. We need to develop and keep the ability to get ahead of the criminals and terrorists and block their moves before they happen. This takes good intelligence and good science. In the end we can out-spend the bad guys like we did the USSR during the Cold War and have a secure border and secure nation.

BTW, I'm not ignoring the East and West Coasts and Northern Border. Our 1980s study didn't address these areas in detail. Many of the principles I mentioned above would need to be applied to the other borders as the threat dictated. Certainly the West Coast will need security to prevent criminals and terrorists from end running the Southern Border defenses. The defense of the East Coast is equally important because blocking off access through the Gulf of Mexico may force the criminals and terrorists the move up the East Coast to gain access to the US. Fortunately, we have a strong ally on our Northern Border. We probably don't want to build a fortress-type land border between us and Canada. We need to work with Canada to prevent the criminals and terrorists from entering their country in the first place.

Place our forces about 2-5 miles back on our side of the border. Then declare the zone from the border to where friendly forces are, to be a 24x7 live fire/bombing range. Post signs at regular intervals explaining such, and that entry into this zone will likely result in death and/or dismemberment. It's a win-win-- the border is patrolled and our forces get a huge new free-fire and bombing range.

When I stopped putting bird feeders in my back yard the birds and squirrels stopped coming over the fence after them. Stop providing the care and start expelling them as fast as we find them. life saving medical care only. Disband the border patrol, make them park rangers and give the border security to the National Guard. let all 50 states work together supporting it.

How would president Don Murphy seal the border? Ooooo - easy one. At the bottom of every I-9 form is a blurb saying that you understand that your employee is legal to work in the USA and a further chunk of the blurb says what will happen to you if you don't check. So basically I'd do spot checks. Got caught? Throw you in jail, disband your company and fire all of your legal employee's. Do not allow the legal employees to sue you for jeopardizing their careers, but advertise where you lived. "Today in BFE, employer Bob Jones' - who lives at 1616 Mockingbird Lane - had his company disbanded due to hiring illegal labor." How many people would you have to do this to? Start advertising HUGE rewards for whistle blowers. Publisize them! "Yeah, my boss was hiring illegal work and I got a million dollars for turning him in!" Soon, its just not good sense to hire them and - OH HEY - they stop coming.

Nothing can be absolutely stopped, as we all know in accomplishing our missions ... "if there is will, then there will be a way".
But besides changing our civil policies (prosecuting employers of illegals, stopping the social dole, etc), the actual administration of the border should be a military function, with the current border patrol acting as a seconded unit for the law enforcement portion of any captured border breakers. This also gives a practical separation to prevent confusion on the Posse Comitatus Act stipulations.
The actual rotation of units on the border really isn't as difficult as it sounds... look at the current map and where our military installations are -- Ft Sam, Ft Hood, Ft Bliss, Arizona Proving Grounds, 7 Palms ... all close enough to provide fortnight company size rotations to each 10-mile segment. As we build guard towers we also build some rudimentary barracks/DFACs for the guard force. Instead of playing "field maneuvers" out beyond visual range of the barracks at Ft Hood the units would get practical field time fulfilling their oaths and "defending against all enemies foreign and domestic".

With current forces? Logistically impossible. The length of our border makes it non-feasible.

There's 5k Miles of US-Canada border (including 1500 miles along Alaska). There's 1900+ miles of US-Mexico border.

We're looking at 7000~ miles of border, not including Ocean border.

If you do a border tower every 500 feet (11 towers per mile), you would need 77,000 towers. Now take that and multiply that by 2 for two people per tower (154000). Three shifts per day would end up at 462,000.

That's without any logistical support. Just guards. Using a "liberal" estimate of 4:1, that would require 1.8M people just for border guards. A more reasonable estimate is closer to 15:1 however (6.9M).

I have long been of the opinion that the VAST majority of US troops should be brought home and engaged in border patrol ops If it's good enough for the DMZ in Korea why not at home? A QRF with coms to sizemic detection wouldn't hurt either.

We have to stop looking at this as solely a military project. Illegals come here because they know that the Government will take care of them. This action has to stop. The Border Patrol and border state agencies need to be on the border turning people away before they even cross. Intense political pressure needs to be put on Mexico to enforce their own laws, instead of providing a highway for people to cross to come to the US. Businesses need to be heavily penalized for hiring illegals. Current laws need to be enforced. If Mexico doesn't want to comply, tariffs or some other means can be used to make Mexican made products less desirable for business to market in the US. We do have the power to put great economic pressure on Mexico.

Mexico as well as Central and South American governments need pressure put on them to end the corruption and drug cartel influence in their countries. Although the US has many corrupt officials, these governments have many more. If the countries were free-er, economically, fewer people would want to leave.

Since several states have legalized marijuana, cartel profits have been cut by about 50%. While I am not a legal marijuana supporter, it demonstrates pressure that can be placed on these countries. It should be looked at for more ideas on how to place economic pressure on them.

Military action should be reserved for those areas of the border that the Border Patrol, State and local officials have deemed too dangerous to patrol. The military could clean these up in short order and turn them back over to local control. Assistance can then be given to the locals to keep these areas safe for Americans.

There cannot be solely a military/enforcement component to this plan. It will take all branches of Government, State and local included, to apply the political and economic pressure necessary to make fewer people to want to come here. The US is not completely socialist, at least not yet. We do possess the economic might necessary to do this.

Illegal immigrants are coming here for work. Enforce the law. Penalize the companies hiring them and the problem goes away. We have been getting better at it, but there are powerful forces at work that don't want the laws enforced. All this talk of a wall is foolish. Deal with the endgame, not the symptons......