The US wouldn't like that. They want Russia out of the Black Sea, which would also make it hard for Russia to operate in the Mediterranean and the Middle East (so they can't support Syria and Iran). Maybe even have a US Navy base in Sevastopol and nukes and anti-ballistic missile systems on the Ukrainian-Russian border. It's been the goal of the US to encircle Russia since the end of the Cold War.
I don't know that they would need a base that close, with ICBM's and stealth bombers there's really nowhere in the world that can't be reached by nukes.
If you have nukes on the border you could do a decapitation strike. Wipe out all command and control sites (ie: nuke Moscow) and fixed missile launch sites before the Russians have time to launch their missiles. The closer the nukes are to their target the quicker you can strike.
This is also why the US didn't like to have Russian missiles in Cuba, which resulted in the Cuban Missile Crisis in October 1962.
There are military hardliners in the Pentagon who believe in a winnable nuclear war. Of course then there are still mobile launch units (of which the Russians have a *LOT*) and ballistic missile subs which could launch missiles on their own in retaliation.