I try my best to reply to emails as soon as I can, because I've learned that it's a good reflection of professionalism. I've interacted with people who reply to emails immediately, and personally, I think it gives off a very positive vibe. Prompt responses will make the receiver more inclined to put more effort into the relationship. It will also make you look like you are on top of things.
Here's a simple tip when you're overwhelmed with emails: flag the emails that need responses when you're doing the routine skimming through your inbox. That way, you'll remember to attend to them when you're done reading your emails for the day. Do you have any other professional tips you want to share?