The simplest non-radioactive method to visualize hybridization is a non-radioactive probe method. If you have ever done a Southern blot or dot blot, you will have completed about 1/3 of this. If you have done a western blot, another 1/3.
The first difference is in how you prepare the probe. The probe is made or labeled with flags instead of 32P. There are three major ways of doing that: 1) purchase a flagged probe; 2) PCR amplify the probe sequence with flag-dUTP instead of dTTP and 3) use Klenow to "run off" labeled probe using flag-dUTP instead of dTTP. The last two are available as kits for labelling from several manufacturers. Three flags are digoxigenin, fluroscein or biotin.
Perhaps you have guessed the detection system. Once you have completed your hybridization, you proceed to probe detection using antibody- or streptavidin-enzyme conjugates like horse radish peroxidase or alkaline phosphatase. The whole detection system using these conjugates is very much like doing a western blot. Again kits are available. Conjugates then are anti-dig-HRP, anti-fluroscein-AP, or SA-HRP as examples.
In brief, the protocol proceeds like this:
Label the probe: (I have used both PCR and Klenow). Set up the reaction, once it is complete, you should check the flag incorporation much like you do 32P incorporation by doing a dot blot of titrated probe. Proceed from this to the detection discussed below.
Once you have sufficient labelled probe, do your electrophoresis and then transfer the DNA to nylon membrane using the method of Southern. Paper towels are pretty minimal technology and we still use it in our lab.
Once the DNA has transfered (over night), let the membrane dry and proceed to hybridization or store in the dark.
Hybriziation: Add sufficent hybridization solution (without probe) to the membrane to pre-hyb for about an hour, then add the amount of probe needed based on the incorporation of biotin. Hybridize with mixing at the right temp for your probe overnight. Next day, wash the blot with required number of washes.
Detection: Put the blot into protein-block solution (skim milk for example). Block with rocking for 1/2 hour or so at RT. Add the required amount of enzyme conjugate for the surface area of the blot and volume of block you have (eg: 1/3000 dil in 10 mL). Rock for 1 hour or so. Wash with buffered saline. Detect with enzyme membrane substrate. Detection substrate can be chromogenic like enhanced mTMB or even luminescent/film for greater sensitivity.
These protocols will take about 3 days to complete and will be faster and likely more sensitive than 32P just because you don't have to wait days for the exposure of the film. Using the chromogenic substrate we have had good results within 10 min or so. With luminescent substrates, as little as a few seconds can be sufficient to get signal on xray film. The enzyme substrate system then depends on your access to darkrooms and developing equipment.
If you google "non-radioactive Southern", you should find links to protocols and manufacturers.