While there seems to be a good amount of discussion about the border wall desired by President Trump, I have not seen much in the way of discussion regarding state-based approaches; that is, given the current state of affairs in the US in regard to support for a full wall, why is there not more of an effort to allow border states to construct their own walls if so desired? It seems some politicians in Texas have considered the possibility, but still via reimbursement by the federal government (https://www.kxan.com/news/local/austin/while-shutdown-continues-texas-state-leaders-mull-funding-border-wall/1698160867); is it due to the expense of building such a wall even for a single state's border being too costly for most southern border states, are there federal laws that would prevent states taking that course of action themselves, or is there simply not enough support in most border states for that to be a viable course of action for those who desire a full border wall?
It is essentially not allowed.
States may not usurp the federal power over immigration. State attempts to regulate concurrently in a field already occupied by a federal statute have been struck down under the doctrine of preemption. In Hines v. Davidowitz (1941), for example, the Court held that the Federal Alien Registration Act preempted Pennsylvania alien registration provisions. Under the preemption doctrine, federal law in a specific area may even preclude consistent state regulations.
States cannot enforce their own borders. This was somewhat recently showcased with Arizona SB 1070 and with Joe Arpaio attempting to control illegal immigration in Arizona. There several other examples of this too, but it comes down to federal preemption.
In addition to the issue David S identifies:
Cost. Texas has a state budget that runs about $108B/year. Cost estimates for a border wall vary a lot, but the most recent request to Congress from the Office of Management and Budget works out to ~$24.4 million per mile. Even if the state decides it only needs a few hundred miles of wall construction (perhaps because it believes the existing fencing and natural barriers are sufficient for the rest), the cost estimate (assuming such a large project is on-budget) would still be 5% or more of the state's annual budget. That's a substantial, though not impossible, amount of resources for a state to commit, and there would need to be popular support to raise the money through taxation or cuts to services; wall supporters would have to put their money where their mouths are.
Maintenance. The structure will need to be maintained. Damaged portions replaced, debris cleared, flooding prevented, gates and locks opened/closed, etc... The state doesn't want to be on the hook for that cost in perpetuity. Would they transfer ownership of the wall to the federal government after construction or would the federal government pay for the maintenance of a state-owned wall? Either way, there'd need to be a legal mechanism to do that.
Regulations. Big long walls violate all sorts of regulations. They interfere with endangered species, drainage, bird migration, etc... A state can't just build a wall inside of parks, the National Butterfly Center, and wildlife refuges. The federal government has the authority to issue itself certain waivers of these regulations for the purpose of building barriers at the border. Those federal authorities wouldn't automatically extend to a state that wanted to build its own barrier.
Land acquisition. The government needs to acquire the necessary land, often from unwilling owners. This is likely to cost yet more money and will result in lawsuits from owners who do not have any interest in selling, lawsuits the state would have to fight in court. Other owners may support the wall, but the practical problems of people being stuck on the "wrong side" will have to be addressed. And those are simpler cases. Portions of the border are on tribal land, and the federal government has certain powers in that regard that states do not. Other portions are on federal land. The Boundary Treaty of 1970 prohibits construction of barriers in the Rio Grande's floodplain without the permission of both countries, something that requires the federal government.
Public support. Politicians along the border do not uniformly support building a wall. Rep. Will Hurd, a Republican, has a district that covers 820 miles of the border in Texas. He's repeatedly opposed construction of a border wall and won the GOP primary with 80% of the vote (the general was quite a close race, but the Democratic candidate didn't support a wall either). Some landowners with property near the border are staunchly opposed, and Texas lawmakers, including Republicans, have noted the degree of opposition. Forcing the issue of building a state wall could be a risk for representatives of border areas that they might lose their next elections. 53% of Texan voters opposed building a wall in a Qunnipiac poll conducted last April, and national polls show majority opposition to a wall. Without broad public support or intense local support, there's little appetite for politicians to stick their neck out for such a project.