It's all going to depend on the post. When we were in COS, it made more sense for us to live off post, bigger house, cheaper to buy, nicer area but they just built a ton of nice housing on post there. In DC area, it's pretty much a toss up for lower ranks. Officer ranks you can totally get a house for under BAH, our townhome is on a 15 year mortgage, is 10 minutes to DHs job, in a top rated school district in the country, and BAH covers mortgage, insurance, HOA (which is trash and snow removal), water, electric and cable/internet. Utilities are NOT free at every post, it all goes off of useage so if you are in AZ and keep your AC at 60, you're gonna be getting a hefty bill, stay under 'standard' usage and you probably won't have to pay.
We never had great experiences living on post in the US, a lot of irresponsible parenting which drove me NUTS (seriously toddlers wandering in the streets no parent in sight), a lot of constantly having to watch what you do and say and to whom you do it with and say it too. You do have the comradery of a military community and having access to pools, gym, commissary etc though so only you can really make that decision.