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

No and a wall is too expensive. Just need our house and senate to pass practical laws and enforce the ones we have. We need immigrants.

There is no way to seal any border, due to the fact that one human invents a seal and many humans invent a way in; the rule of "Necessity is the Mother of invention" applies both ways.

I think the goal is to slow the flow... the wall has to happen.

The physical land borders are the easy target for anyone wanting to gain entry and for politicians to place blame on the issue. The other entry points, by air and sea are never discussed. Millions of non citizens enter this country on work visas, student visas, tourist visas, entrepreneurs visas (the Chinese currently have a dominance on the 10,000 total per year). and let's not forget the second most popular way to gain entry, child birth. All airports are birthing centers, the Chinese, Phillipines, India, all the ME, Eastern Europe use our airports as birthing centers to grant that newborn US citizenship. China uses US territories in the Pacific for this, thousands of American born Chinese are now in China, with a birth certificate, and once they are of age they can legally enter the US with family. So are the N/S borders a major problem, well yes. But the Immigration Laws are the real problem.

You had me at "minefield". No border, no sovereignty, no sovereignty, no country. Enforce the law. Strike at the mules in some way so draconian no one would want to be a mule. Enforce the law, as SGM Marquez says.

Tom Kratman once pointed out in one of his articles that for every 50-foot wall, there's a 51-foot ladder, but also that for every 51-foot ladder, there's a 50-caliber.

It would be a logistical nightmare, though. It would also require an extreme amount of political capital and the willingness to persevere despite the howling through both diplomatic channels and through pro-illegal immigration groups in the States.

The issue is thorny, because it involves people who are often destitute. Humans are hardwired to help the destitute.

A militarized border could be done, but it really ought to go hand-in-hand with a streamlined immigration process. My wife emigrated from Canada, and the process was a total cluster from the very first form we filed. Not to mention expensive. I had to file a form declaring my financial ability to keep my wife off the dole, and the cost to file that one form was ~$400-ish.

We are never going to solve the problem of people wanting to come here, so long as our nation remains the economic powerhouse of the world. What we have is simply too enviable.

That said, we should balance inviting folks to live here with ensuring that they're fully integrated into our society, both from a legal standpoint and an ethical one. Diversity can be a good thing, but we're greater than the sum of our parts, and it's time we stop being fixated on what wonderful parts we are and start focusing on becoming that greater thing.

Sure it can. But to completely seal our borders:
-it would most likely be conducted by the military; Army on land, navy and coast guard on the seas and shores, and all branches in the air, both jet aircraft and helo.
-to provide for such a force would take several tens of thousands of people.
-the cost of the forces, equipment (radar, vehicles, etc), and support (food, fuel, housing, etc), would be so astronomical that it would never get out of committee when proposed to congress.

The term "reasonable" is relative. For example, some people see $600 billion in military spending as reasonable and others think it's a waste. It depends on perception and personal biases. But I don't want to get caught up in semantics. Perhaps the answer is NOT sealing the borders but rather a multi-pronged approach; an approach that places security on the border and increases the risk over reward for those attempting to cross and those who would help them.
Maybe a wall is feasible in some sense. No so much from the Gulf to the Pacific in the south and ocean to ocean in the north, but perhaps at strategic points to redirect border crossers, much in the way of tank ditches, wire, and minefields in conventional warfare. You back up the wall with troops from existing units. Put on 30 or 60 day rotations at designated border camps (to be built); you could maybe even involve the National Guard and Reserves, incorporate their 2 week exercise on border duty. The objective of the military is primarily detection and apprehension. After apprehension, they are turned over to the border patrol for investigation, processing, and deportation. In this manner, it could be argued that the military is NOT enforcing law but merely upholding it's primary purpose which is defending a country. Much the same way we conducted border operations in Germany before unification. Yes, I realize that was more like patrolling the borders of Oregon than a country the size of the US, but you got to start somewhere.
After "securing" the border, you work inwards; this is where federal, state, county, and local agencies come in and enforce existing law that states it is illegal to aid, transport, harbor, and hire illegal aliens. This means not just going after the illegals themselves, but also those that hire them. If you want imprisonment as part of punishment for the employer, you can do that. I think what's bigger is to fine them heavily for each and every illegal they hire. Make the fine hefty enough that it becomes to much of a risk to hire the illegals. Then give them an option. They can pay the fine and let them be deported, or pay the fine then sponsor the illegal and obtain a work visa. While taking down employers, every illegal caught speeding, jaywalking, loitering, or however you catch them, gets deported. There ill be no need to round up the 20 million people here illegally. You simply make it uncomfortable for them to be here and the flow will slow and some of those here may even leave on their own.
Yes, it's a big cost up front but this should begin to taper off before too long.

It always surprises me that otherwise intelligent people are so mesmerized by the news media and our political parties that only two solutions are envisioned to the immigration issue. (i.e. Completely open border and citizenship for illegals versus a tightly sealed border.)

May I suggest an alternative? We need a guest worker program, because you don't need to be a citizen to work in the U.S. nor do you need to be a citizen to be paid a fair wage. We could issue ID cards with encoded biometric data. You want to come in, you show you ID and scan your fingerprint.

Side benefits:
1) If violating the law carried with it the possible permanent loss of your ID, there might be more respect for U.S. law, including liability insurance for alien drivers.
2) Easy legal entry would reduce the number of people seeking illegal entry. This would make terrorist, drug runners, and other smugglers easier to spot.
3) With less "innocent" people crossing the border, more extreme measures could be taken against criminals, with less outcry.

Maybe someone needs to talk to Israel, they've seemed to do well on this subject. Yes it can be sealed, but you need over watch on the obstacle. As SGM Erik Marquez stated in his post, there are many things which can be done to take away the incentive to come.

Build to fences, One on the souther border. The second fence is enforcement of immigration law, eliminating social services, housing, schools, government ID, housing and jobs for illegals and they will climb back over the fence to go home.