How do I stop my Yahoo account from continuously downloading emails?

Launch Mail, select "settings" (drop-down under "Mail"), and in the window that opens select "general", and second line down is "check for new messages", and pick the frequency you want. I select manually, but if you are a business you would want a frequent check and there are many choices.

another point: Yahoo apparently thinks you have a new computer and is bringing you up to date of what is on their server. Is it possible you have never deleted any inbox messages on a previous computer? Also, all the sents will download to a new computer. Just let it run, delete as appropriate afterwards.