Mining the Howling Mine
In the first release of the game there were 3 different types of artifacts: Poly, Mono, and Continuous. Poly Artifacts were artifacts that had activated abilities that could be used without tapping the artifact. Mono Artifacts were artifacts that required tapping to activate their activated ability. And Continuous Artifacts don't have activated abilities, however their abilities stop working when they are tapped.

Howling Mine was printed in Alpha as a Continuous Artifact, so its static ability didn't have an effect if it was tapped. The rules worked like this until 6th Edition, and at that time the rule was mostly just dropped, except for a few cards that received errata to maintain that functionality. Howling Mine was one of those cards, but there were a few others like Winter Orb.

One-drops that can be repeatedly used to tap them would include Trickster Mage and Vedalken Certarch. And yes, you draw one card for each Mine you have on the battlefield, so with 4 Mines you would draw 5 total cards.

Aside: with the right deck, you don't need to worry about your opponent having 4 extra draws / turn.  An example of this, is turbo fog.