V.J.
It's important to have your dog examined by a veterinarian, vaccinated, and determined to be in good health. You might also do a brucellosis test to make sure he is not a carrier-- http://www.peteducation.com/article.cfm?cls=2&cat=155...
A certain percentage of Beagles have hip defects so it would also be a good idea to have his hips x-rayed (whether or not you choose to have them evaluated by the OFA-- http://www.offa.org/stats.html#breed ). You can charge a higher stud fee for having documented proof of his hip health plus his AKC registration. If your boy has any abnormalities, it would be responsible to have him neutered to prevent any litters inheriting a defect.
If everything checks out well, you could try placing an ad in the local newspaper or in Dog Fancy Magazine (printed or online)-- http://www.dogchannel.com/classifieds/dog-breeder-directo...
Once you have found a mate for your dog, having a written contract with the owners of the female will prevent any disputes that could occur-- http://www.akc.org/breeders/resources/guide_to_breeding_y... . Both parties should sign and receive a copy. You might want to verify that they have information on her as well. Even if you choose not to breed to another AKC dog, you should make sure she is vaccinated to protect your own dog. Here's a sample stud contract-- http://www.pinnaclepetsupply.com/studcon.htm
As far as the actual breeding, some people choose to have the dogs stay together at one of the owners' homes for the duration of the breeding process. This is more typical of larger-scale breeders with kenneling facilities. Others prefer to meet up a couple of times to let the dogs do their business and then take their dogs home separately. Unless you know the people well and trust them, I'd probably do the latter. Unfortunately, not everyone is honest so just use your instincts and if something doesn't seem right with the owners of the female, then I wouldn't get involved with them. It could turn out to be a massive headache.
I've been a veterinary technician for 9 yrs and my parents used to breed AKC German Shepherds. If you have any more questions, I'm happy to help. Best of luck! :)