It would be hard to tell without pictures or a model number. The model number is probably going to be odd. It will have a three-digit number followed by a decimal point and three to seven more digits. (516.891 or 158.18033 or something like that. The number is usually on a plate on the front of the machine under the bed - where it's hidden by the base or the cabinet when the machine is set up for sewing.
The oldest Kenmore I have is (I think!) a 1959 - it's a Lady Kenmore 89 with all the bells and whistles and runs perfectly. It's one of the few Kenmores that's considered "collectible," although certainly not the only beautiful one that was put out by Sears. (love that picture of the PINK one - I'd buy that one in a heartbeat!)
Kenmore did usually put out a good, solid machine, but there was one group (and I'm sorry, I don't remember which one it was) that is nothing but trouble. I think the prefix was 117 for those machines.
The biggest trouble is finding parts if there is a part that's worn or broken. Sears isn't carrying much in the way of replacement parts. I think that's partly why Singers are collected so often - replacement parts are pretty easy to find for all the old household Singers.