Fretting just behind the fret requires the least amount of effort from your hand so try to fret there. You will need more pressure compared to guitar just due to string thickness and tension.
Getting your bass setup properly is a good thing to do if you don't know how yourself. I personally play with fingers and keep the strings as low as I can before I get fret buzz. So keeping the attack low gives you lighter touch (perhaps easier with them weak guitard fingers

).
Also I suggest using your pad instead of tip to fret strings. This way you mute the strings nicely. Most of the time you'll be playing just one string at a time so that helps with the harmonics problem. Just rest your fingers on the strings and press when you need to.