My son is in Meridian for A-school - he purchased internet for his laptop but he says it's not great but that it was reasonably priced. He hasn't said anything about cellphone coverage - he has Straighttalk which I think uses Verizon. Pretty sure they have an area with internet access on base for the Sailors.
But I think limited reception may just be one of those things they'll have to deal with depending on where they end up
There is no free wi-fi in most A school barracks. In some cases, they can purchase internet service from a local provider, but they pay for it themselves, usually split among roommates. Free wifi is usually at the local liberty center or USO on base, or someplace like McDonalds.
But yes, bad reception in Navy buildings is a thing seemingly nationwide. At least he is in Florida, where going outside the barracks to make a call is nice and warm.
If he finds a solution, please share it as the info may be of use to the next person asking about wifi in Pensacola.