There is a style of dog collar, nylon web straps that fit around the neck and around the ribs with two straps (one down the back, one down the chest) with a leash ring on the back. The name of this type escapes me but any large pet shop would have them. Attach the leash as you normally would and either hold it (not a good idea for a large dog) or attach to a seat back brace or some other strong point.
Give her enough slack to sit comfortably but not enough to get past the edge of the cart. Is she spooks she cannot jump out, in an accident or emergency move she cannot be thrown out and the strongest part of her body is securing her.
This website has something like a seat belt for dogs:
Ruff Rider Roadie - Canine Vehicle Restraint
Also, Petsmart has several items that could be adapted. I didn't see anything specifically for a golf cart but you might want to ask - if there is a market someone will make a product.
http://www.petsmart.com/family/index...goryId=3245635