A 48/32 subcompact was practically made for this.
I'm using an 8 speed triple crankset with a 10 speed drivetrain with no issues. The concern is generally about front shifting and total takeup on your rear derailleur. It looks like your bike has stem shifters, which I assume are friction shift. If so, can use basically any rear derailleur you want if it'll bolt up to your hanger, if somehow your current one is insufficient for whatever range you need. A longer cage rear derailleur is probably a good idea.
Whatever you do and whatever you change to, make certain that your chain is the correct length for the large large and small small combinations. Especially if your chain is too short, it could do a great deal of damage to your drivetrain and/or frame if it doesn't fit! A new chain is <$20, so don't take the risk.