From what I've read a trampoline is like an above ground pool. Even if I didn't put it up I'm potentially liable for injuries since it's on my property.
From
https://www.chicagotribune.com/real-estate/ct-xpm-2010-04-04-ct-mre-0404-renting-trampoline-liability-20100404-story.html :
"unlike other types of portable items, such as patio furniture or potted plants, a trampoline can be dangerous. A landlord generally has a duty to keep the property reasonably safe for tenants, their guests and any other legitimate visitor (as well as any pint-size curious visitors too — more on that below). This is why the landlord must fix loose stairs, make sure vents work properly, keep the electrical system safe and so on.
You cannot absolve the landlord of these duties. In legalese, you cannot waive your right to a reasonably safe place to live.
For instance, you can't say, "No thanks," when the landlord insists on replacing that retro wood stove that has vents that are prone to clogging, risking carbon monoxide buildup, no matter how inconvenient or disappointing it might be to lose that stove. Nor can you say that you'll keep it and take your lumps if and when the stove causes a problem.
...
In light of these risks, your landlord has correctly identified your backyard trampoline as a "dangerous condition" on his property. He may be worried about his own liability should one of your children be hurt. It would take some cheek on your part, to be sure, but it's not entirely inconceivable that you could file a claim against the landlord's liability policy if your child is injured on the trampoline, asserting that the landlord bears some responsibility for the injury because he allowed the trampoline in the first place.
And suppose a child from the neighborhood discovers the trampoline, uses it without adequate supervision and gets hurt? Here, the landlord's chances of being liable go way up because he might be seen as permitting an "attractive nuisance," a dangerous condition, like an unfenced swimming pool, that is particularly likely to appeal to a child."
From what I've been able to ascertain it's rare that renters' insurance would cover liability for a trampoline.