Purchase an electronic tuner. While it's possible to tune by ear, most beginners have a tough time determining the difference between one note and another--which is an essential skill for properly tuning a guitar. There are many electronic tuners available, so make sure to get one that is specifically recommended for tuning bass guitars.
Turn on your electronic tuner and hold it up to your bass guitar, right by where you usually pick the strings at. Pick the low e string (the thickest string) and look at the electronic tuner. If the string is in tune, it will say "E"; if the string is not in tune, it may say another letter. If the tuner says the string is currently tuned to F, you'll want to tune down, but if the string is tuned to D, you'll need to tune up.
Adjust the tuning peg on the low e string according to what the tuner reads. To tune up, turn the e string tuning peg left. To tune down, simply turn the tuning peg to the right. When turning the peg, make sure to do it slowly to prevent the string from breaking if you go too far.
Re-check the e string with the tuner to ensure it's in tune after you've turned the tuning peg. Adjust the tuning peg as necessary until the electronic tuner says the string is tuned to e.
Pluck the A string (next string down from e) and hold the tuner up to the string. Look at what the tuner says. If it says your string is in B tuning, you'll need to tune down. However, if it says the string is tuned to G or a lower note, you'll want to tune up.
Turn the tuning peg to either the left (to increase tension and tune up) or to the right (to decrease tension and tune down) according to what you've found from using the tuner. Re-check with the tuner after you've done a bit of adjusting until the string registers as being tuned to A.
Hit the D string (next string down from A) and allow the electronic tuner to read the note. If the note that is read is C, you'll want to tune the string up; if the note registers as being E, you'll definitely want to tune the string down so that you can get the bass guitar into standard tuning.
Adjust the D tuning peg according to what the tuner has told you. Tune it down or tune it up and re-check after a bit of adjusting. If your string is very out of tune, you might have to spend a few minutes adjusting until you get a perfect D note to register.
Play the G (thinnest) string openly and use the tuner to determine what note has been played. If the note registers as a G, the string is in tune. If not, you'll need to either tune the G string down or up, depending on if the note is a A or F. Take your time in adjusting the string, as the worst thing to do here is to adjust so fast that the string breaks.