I always hitch the trailer to the bike first. I lean the bike against something and load our 16 month old. Our 3 year old is able to get in on their own. The 3 year old does not get in until the baby is in. If there is nothing to lean the bike against I try to find a slight slope and lay the bike pointing up the hill to load them.
When I load with the bike standing, I keep my leg against the trailer bar to keep the bike from tipping over. Of course, with the bike on the ground, there is no danger of tipping over.
I am guessing that your problem is a result of your handlebars. If you lean a bike against a wall, longish flat handlebars will cause the front tire to be turned off center. Typically the wheel will point away from the wall. In this configuration, the bike is much more likely to fall as the bike is given a way to roll away from the wall. Corners of buildings work well to help overcome this if you put the handlebars so that they hook the corner. Barring that, you would need a wall that is lower than your handlebars to get a stable lean.