One of my doors has these gaps on the bottom and side panels. Is there a way I can fix/fill them somehow?
Adding more photos below:
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It appears that the trim wasn't properly toed into the door frame. However, it's also possible that there is drywall or a 2x that's proud of the doorframe.
You can fix the gap with painters caulk. However, I would pull the trim off and determine what is causing the gap. If it's drywall it should be easy enough to cut out to fit the trim properly. If it's framing causing it you can plane it down. From there the trim shouldn't be an issue.
Some really good points being made by contributors on possible structural issues. Some questions:
These are all indicators of possible structural shifts that should be assessed by a professional through an on-site inspection and are probably beyond the scope of what we can tell you here.
Something isn't right if I understand the picture correctly. I see that there used to be caulking that filled the gap and now the gap is wider. This means something is moving. Moving things in a structure is not a good thing. You need to find the source of the issue to be able to address it.
Here is the first thing that I would check. Note that the door opens left. This causes a force to the right on the lower hinge. It's quite possible that the connection of the door frame is inadequate or has failed which is forcing the door frame to move to the right. A lot of times this connection is just a nail. A way to test to see if this is the issue is to open the door and carefully lift up and then push down on the door handle, if there is movement of the bottom of the door frame at the wall, this is probably the issue. Also, the gap will be larger at the bottom of the door than then top.