<p>I'm having an issue where the first layer of my support structures isn't sticking on the edges and causing the print to (eventually) fail.  Any ideas on how I can fix it?</p>

<p>I'm using Slic3r.</p>

<p><a href="https://i.stack.imgur.com/XD4AX.jpg" rel="nofollow noreferrer" title="Photograph of first printed layer"><img src="https://i.stack.imgur.com/XD4AX.jpg" alt="Photograph of first printed layer" title="Photograph of first printed layer"></a></p>

There are many things you'll need to check and/or confirm to ensure that you will have a good bond to the bed. The first is to confirm that you are using a genuine Prusa printer as it appears in the photo. Having built one recently makes it easier for me to guess that is the case.
Have you performed the bed calibration sequence? The manual provides a series of steps which results in a zig-zag pattern of filament being placed on the bed, while the z-height is adjusted from the panel. You want to have a filament trace that is only slightly squished onto the bed, not flattened so much that it's cutting into the PEI and not so high that it's nearly cylindrical.
The bed must be of the correct temperature for the filament selected. If in doubt, raise it five to ten degrees C. I recently assisted with the aforementioned printer that had a peeling problem and the bed temperature had to be raised to 70°C from the "standard" 55°C generated by Slic3r.
It is critical that the bed be clean as well. Denatured alcohol is recommended, with application of a clean cloth.
Your photo is somewhat out of focus, making it difficult to determine if the brim is being created at an excessively high z-level, which will cause peeling. The main body of the print, also out of focus appears to be heavily flattened, but that could be an artifact of the photo.
The reflections on the bed appear to indicate that some gouges in the surface exist. If your PEI is damaged, you will have the problem you described. I've seen videos in which the bed is not quite as gouged and was refreshed with very light sandpaper or very light steel wool or both. Of course, after using such material, clean the surface thoroughly.
I understand the PEI that is applied by the manufacturer is quite thin and can be further damaged if too much pressure is applied while refreshing. It is far better to apply too little pressure if you plan to perform this task.
Consider to read through the manual and address all of the calibration aspects of the printer to establish a base point for the problem you are experiencing.