Using a ladder or any stairs available, reach the top of the roof. Locate the dome and check if there are cracks on it. Use a ruler to measure the length and the width of the cracks on the dome. Make sure you check all the cracks or any other damage on it.
Image courtesy: deanwolf.net
Take a caulk gun and fill it with tube of silicone sealant. This will be used the fill the gaps in the skylight dome. Now place the tip of the caulk gun to the edge of the first crack and press it downwards to release the silicon the cracked area. Now move the gun along the whole crack and make sure the crack is filled with the silicon.
Image courtesy: homedepot.com
Repeat the process for each crack in the skylight and allow it to get dry for around 1 hour.
Remove the all weather tape from the dome and take a new tape that you will apply on it. Cut a piece of the tape, which should be slightly bigger than the first seam. Place the piece of the all weather tape on the first seam and drag it down, in order to adhere it over the seam.
Image courtesy: tapes-direct.co.uk
Repeat the process of placing the all weather tape on all the seams on the skylight dome. Now check the bottom corners of the dome, which connects the dome to the roof. Check if there are any cracks. If you find some, repair them using the same procedure which you used to repair the outer part of the dome.