Yes, many were drafted into the military for the Korean War, and someone born in 1933 would have fallen into the correct age bracket.The National Archives has military records.Try this link for addresses and info on ordering a copy of military records.If there is a military record, it should say what the enlisted person's home address was, as well as where he mustered out.Those two pieces of information are decent places to start looking, though it's no guarantee he would still be there.From there you can try city directories for addresses, etc.Also, you can try looking for web sites or membership groups of any unit he was assigned to, that might get you current info as well if he belongs/belonged to or registered with any of them.Also, if you just search Korean War records, you'll come up with a ton of sites that might be helpful.
Contact the National Personnel Records Center at: Civilian Records Facility 111 Winnebago Street St. Louis, Missouri 63118-4199
Military Records Facility 9700 Page Avenue St. Louis Missouri 63132-5100 Phone: (800) 318-5298
At minimum, please include the service member's complete name, social security number and/or serial number, and the requester's return address. Website is: http://www.nara.gov/regional/stlouis.htmlhttp://www.nara.gov/regional/stlouis.html
If the cousin is deceased, his social security number would be listed on any of the genealogy sites with the Social Security Death Index, such as ancestry.com or rootsweb.com