Hopefully you get this reply before you have taken your oath!
Here are some options that you might want to consider before enlisting in the military. Since you already have your degree why not apply for an officer program? They get paid ALOT more... always live off the ships... and their job is more paperwork related and some leadership (depending on the path you go).
You go to OCS (officer candidate school) commonly known as spoon and fork school. Once you go there you get to run and exercise and pretty much learn everything about the military in 12 weeks. Sort of like boot camp for officers. Once you graduate you are commissioned an ensign. You get paid O-1 pay which is about 2.5 times higher then what you would get as E-3 pay.
Downfalls.... well you wont get into the Navy Nuke program as an officer. The degree requirements are you need a technical degree with 1 year of Calculus based Physics and 1 year of Calculus (CALC 1 and 2) along with great SAT scores or ACT scores.
Good things .... Officer pay, officer respect (when gained) you live off base all the time, you advance to O-3 (LT) within 4 years.
If you have your degree there is absolutly no reason you should ever go enlisted unless you are dying to go through the nuke program to become one of us

I am a current nuke instructer at NFAS (nuclear field A school) here in Charleston SC. It is a GREAT program and I am not just saying that.
Key things to remember when you get down here if that is the path you choose.
1. The nuclear program is NOT like college at all. It requires ALOT of effort, alot of time, and more importantly alot of self motivation to do well.
2. Dont believe everything your recruiter says, like a few other people have said down below they are trying to meet a quota.
3. If they try to tell you you dont qualify for an officer program with a degree... go find yourself another recruiter because those are the type of people you dont want to listen to. They just dont want to do the paperwork involved to get you into an officer program plus you have to go to an entirely different recruiter, hence he doesn't make his quota.
4. The Navy really isn't all that hard. If you can listen, do what you are told, and work without complaining you will excel. If you show you want to do well, you will be helped by many many people.
Is the nuke program worth it?
Absolutely 100% yes. I know that when I get out at the end of the year I am pretty much guaranteed a job 65K (to start) and 100K+ a year within few years... I finished my degree, qualified all the watches I could, and listened to others advice (expecially my chiefs) If you have your degree look into the officer programs. Dont listen to the recruiter thinking once you go to MEPS you are in the Navy... The only time you are officially in the Navy is when you take the oath to support and defend this great nation!
Anyhow I commend you on your great decision to enlist/become an officer in the military. You WILL work with the best people in the world and join a brotherhood/sisterhood(politically correct) that you will never forget!
"Any man who may be asked in this century what he did to make his life worthwhile, I think can respond with a good deal of pride and satisfaction, 'I served in the United States Navy,'" wrote President John F. Kennedy in August 1963