Fingering Notes
From any position along the fretboard, your fingers can reach four frets. Usually, bass players use one finger per fret. This can be difficult at the lower, widely spaced frets, but it allows you to play without shifting your hand around very much.
Another option is to only use your first, second and fourth fingers for playing notes, with the third finger doing nothing except aiding the fourth in holding down strings. This is the way notes are fingered on the upright bass. This method is easier on your hand and needs less coordination, but you can't reach as many notes in the same hand position and you have to shift a lot more. You may wish to do this on the lower frets to avoid stretching your hand too much, but use all four fingers higher up.
Left Hand Health
Too much bass playing with the wrong hand technique can lead to pains and possibly injury. The easiest way to hurt yourself is to play with your wrist bent (which often happens when trying to lean forward to look at your fretboard while playing). Cranking it forward to reach notes or having your hand scrunched up against the neck can do damage. Keep your wrist straight.
Your fingers may be a source of discomfort as well. If you notice a lot of soreness in the muscles of your fingers, you might be squeezing too hard. Experiment to find exactly how much pressure you need to exert to allow the fretted string to ring clearly - it doesn't take much force to get the notes sounding good.
You will also get tenderness on your fingers from the strings rubbing against them. This is actually a good sign. It means you are well on your way to a nice set of tough callouses, a point of pride for any serious bass player. If you get blisters, take it easy while they heal.