Should I put a safety gate for the stairs to prevent my baby from accidentally falling or am I being paranoid?

Good timing! It's interesting that the national falls prevention awareness day was just observed two days ago--the first day of fall (sept. 22 in 2012) to promote and increase public awareness about how to prevent falls among older adults. But, the risk of falls among infants is real. Approximately 1 million children are injured in falls each year. Some of these falls are from stairs. Good call!
Put up the gate. You are not being paranoid. Babies can suddenly be crawling and moving faster than you ever expect. Putting up the gate will ensure your baby doesn't fall down the stairs.