I have just moved into a 1920's house where the washer/dryer are in the basement. The drain seems to go into a hole in the wall that drains out to the sidewalk. There is no sump pump. The basement is very damp too. Do I need to connect into the city sewer stack to have it drain properly?

Washer should not drain to the curb. It could have been OK 100 years ago, but definitely not in 2015. It is only matter of time before somebody notices foaming water running down the curb, traces it to your house and call Health Department Inspector on you. You have to connect the drain into house waste water drainage system as any other plumbing fixture in the house.

Milo

Yes you need to connect. I know some people who live on farms and have septic tanks that have theirs run into the yard, but even that is rare today.

If it's your house, you better fix it. If you rent or lease, then call the landlord. It's probably been that way a LONG time.