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.