Considering this is an 88 Civic the answer really would depend on how much time is the car sitting between times of when you put a new battery in the car to the time it has slow starting. If it is not sitting and you are using the car regularly I would first check the brake lights make sure those are OFF when you are not in the car, make sure all doors are shuting properly, Make sure that the radio is not running while the car is off. I had one car that the radio would not shut off properly and was draining the battery, and Check all lights(trunk, dome,&doors.). Unless the Battery light is on then there would not be a problem with the Alt. Especially that considering that you can drive it without the battery dying while driving that is another sign it's not the Alt.If the car is sitting a while between drives then that can drain the battery on it's own just due to the fact that the car is old and needs to be at least started and run for 15-20 min. regularly so if it's sitting for a long time and then starts slow you may want to try starting it once a week and just let it run for a little while to let the Alt. charge the battery a little bit.