There are a lot of alternatives to nuclear weapons which are not explored in movies because the choice would be rather anticlimactic. Remember that these movies rarely make any sense; it's all about shock and awe, not logic.
Consider that a flesh and blood creature is going to be severely damaged by an explosion in real life, unlike movie monsters which are conveniently immune to armor piercing tank shells, napalm, etc.
First things first, this giant creature would be targeted by air strikes, and maybe face ground armor capable of delivering punishing ordinance. Have you ever watched a movie and heard the main character yell into a radio:
"Coordinates X, Y, Z, this is danger close!!!"
That's because the shock wave of an explosion (or the resulting shock wave from the impact of a kinetic projectile) can kill a human being. It's called hydrostatic shock. You never see Godzilla worrying about that in the movies, but in the real world that's something we're quite mindful of.
Also keep in mind that we are experimenting with Gauss cannons which fire square blocks of metal because aerodynamic ammunition would actually go too far. You know it kicks ass when that becomes a concern. And while there's severe limitations to the technology today, that will likely not be the case in 10 years, and I'm sure some exceptions could be made for an emergency situations. Anyway, moving on.
If for some reason regular ordnance proved to be insufficient (perhaps because there are a lot of parasitic monsters following the kaiju, which would difficult to wipe out in street to street fighting), there exist air fuel bombs, and other truly scary weapons which can deployed, and which are capable of wiping entire towns off the map while not being nuclear.
Only if these things failed would nuclear weapons even become an option, and most likely because there's a concern about some sort of contamination which might be killed off by the radiation fallout. To my mind, that would actually be the best reason to deploy nukes in the first place.
Imagine a giant monster rampaging through the streets of a metropolis. Sure, the monster is dangerous, but it's blood may be toxic, and there may be parasites which stream off of it, and which are actually a bigger threat than the Kaiju itself (think Cloverfield).
A nuke would solve that problem nicely. Kills the monster, and irradiates the heck out of any biological material left behind, killing any bacteria or viruses. Of course the collateral damage may be quite high, but it all comes down to a cost-benefit analysis. DO you risk these things getting out of the city and having to hunt them down for the next decade, or do you turn New York to cinders? And of course it's always New York.