4:47 p.m. on February 15, 2013 (EST)
Use brush to help clean the current seams with isopropal alcohol (91%) or the paint thinner. Try a small patch in a not important part of tent to make sure it doesn't react with the tent fabric.
Erect the tent so that it has time to dry - sunny day helps.
Buy some seam sealer and skip to SEAL IT (or keep reading Do It YourSelf)
DIYS
Clean small can (tuna can is perfect). Two might be needed.
1/2" cheap brush, half of the bristles cut off so that the bristles are stiffer. Get two.
GE Silicone II tube (small is better).
Odorless paint thinner.
Fill can 1/4 full of paint thinner.
Put in enough Silicon to fill it half full (50/50 mixture).
Mix until it 'dissolves'. Actually, it becomes a colloid that has about an hour 'pot time'. It will happen, you just need patience and persistent slow stirring.
If you run out, make up another (or partial) batch in a separate clean, dry can and new brush.
Toss the old can and brush.
SEAL IT
On the inside, start near floor and work your way up the seams to top of tent from back to front (or exit) of tent. Hopefully this will keep you from getting the stuff in your hair. A hat for the messy and clumsy is good. A dog or a cat (or another person) inside the tent is not productive,.
With the brush, daub a small amount on especially where the threads go through the fabric and work it in. You can smear it with your finger if you think that a better tool. Clean up with paint thinner.
Have large boxes handy to put inside the tent to keep the sides away from each other. Depending upon how much moisture in the air (humid days work faster) depends upon when it will 'dry'.
Before putting the tent away, turn it inside out to air for awhile too.
You shouldn't have to do both sides unless you are paranoid about your technique. It does add weight.
Is this a single wall tent? Or are you doing just the rain fly?