I'm going to post answers for both hemispheres, because there are good points from both sets of directions I've found.
For the northern hemisphere:
The "declination drift method" is the most accurate way to accomplish this. The method is straightforward, but it does require some time and patience.
- First, aim the mount's polar axis roughly at Polaris. Now point the telescope at a star that's somewhat above the celestial equator and as close to south as you can judge by looking opposite Polaris. Put in a high-power eyepiece. If the eyepiece has cross hairs, center the star on them. Otherwise put the star on the north or south edge of the field and defocus it a little. Turn on the clock drive, and ignore any east-west drift.
- If the star drifts south in the eyepiece, the polar axis is pointing too far east.
- If the star drifts north, the polar axis is too far west.
- Shift the polar axis left or right accordingly, until there is no more drift.
- Now aim at a star that's near the celestial equator low in the eastern sky.
- If the star drifts south, the polar axis points too low.
- If the star drifts north, the polar axis points too high.
- Again, shift the polar axis accordingly.
- Now go back and repeat from the beginning, because each adjustment throws the previous one slightly off. When all visible drift is eliminated the telescope is very accurately aligned, and you can take long deep-sky exposures.
Via Sky & Telescope
For the southern hemisphere:
- Direct your telescope to a bright star that's low-ish (about 20 degrees) on the Eastern Horizon, near the Celestial Equator (i.e. 0 degrees DEC). The Orion nebula would be good to look at in mean time (given the right time of the year - as it may be too high or not visible).
- At this point, the east/west rotation of your mount has little effect, leaving you to correct the angle of elevation of the mount.
- Now, if the star drifts NORTH, your polar axis is too low - change the axis to angle further up in the sky (i.e. the latitude knob on your eq mount)
- But if the star drifts SOUTH, your plar axis is too high - change the axis to angle further down toward land (i.e. the latitude knob on your eq mount).
- You can tell if the star is drifting north or south by allowing the star to drift for a little bit - now, move your telescope to "catch up with the star" using the DEC control. If you see it heading north, then it's drifting north and vice-versa!
- Now, point the telescope to a bright star on the Meridian and on the celestial equator i.e. right above your head somewhere. This is so the elevation of the mount (as corrected in the steps above) will have minumal effect, allowing you to accurately correct the east/west rotation of the mount.
- Now, if the star drifts NORTH, your polar axis is too far EAST - so rotate your mount further West (i.e. rotate your mount clockwise while looking down at it).
- Now, if the star drifts SOUTH, your polar axis is too far WEST - so rotate your mount further East (i.e. rotate your mount anti-clockwise while looking down at it).
- Repeat these two steps a few times to increase the accuracy of your plar alignment - repeating Part 2 will allow you to get Part 1 more accurate.
Edit to provide less-precise-but-more-prompt method
Polar alignment for southern hemisphere observers is very similar to techniques used for the northern hemisphere. The major difference is the lack of a bright pole star like Polaris near the south celestial pole (SCP) to aid orientation used in several alignment procedures.
Rough polar alignment using the latitude scale of your scope is identical to the procedure used in the northern hemisphere.
Either look on a map, use Google Earth, or consult an almanac to find your observing site's latitude. Unlock any latitude adjustment screws on the sides of the mount and turn the latitude adjustment screws until the index on the polar axis reads your latitude. Tighten the adjustment screws if needed to secure the latitude setting. (You may also need to loosen the center pivot bolt by turning the hex nut to allow the equatorial mount head to be tilted.)
Latitude adjustment of the mount
Now complete the polar alignment by turning the entire mount (not either axis; both should be clamped tightly) to align the upwards end of the polar axis with south on the horizon.
Another more accurate rough method is to point to Sigma Octantis. This star is located only one degree from the SCP, the point in the sky around which all the other stars appear to rotate, and where the polar axis of a properly aligned equatorial mount should point.
First, set up the mount so that the polar axis is pointing south.
Second, unlock the declination clamp and move the scope in declination so that the tube is parallel to the polar axis. Your declination setting circles should read 90° in this orientation. Clamp the declination lock.
Note: These last steps involve moving the entire mount. Don't use either the RA or Dec motions to change the position of the tube.
Move the mount in altitude and azimuth until Sigma Octantis is in your finder's field of view or centered in your finderscope. At magnitude 5.5, it is a fainter star--just slightly brighter than the unaided eye limit--so your finder will definitely be helpful here.
Sigma Octantis and the SCP
Tweak the position of the mount by again moving the mount, this time centering Sigma in the eyepiece field of view. Altitude can be adjusted using the latitude adjustment screw or shortening-lengthening tripod legs.
For greater accuracy, you can point directly at the SCP using an offset from Sigma Octantis. (This can be done after aligning the optical axis of your finderscope with the polar axis.) Use the chart above or these patterns of southern circumpolar stars to find the SCP. Draw an imaginary line toward the SCP through the Southern Cross stars Gamma Crucis and Alpha Crucis (down the long axis of the cross). Draw another imaginary line toward the SCP at a right angle to a line connecting Alpha Centauri and Beta Centauri. The intersection of these two imaginary lines will point you close to the south celestial pole.