Have you tried with a new, empty ArcMap document, add the basemap first, then add your point feature class to it?
The first layer added to ArcMap sets the coordinate system of the map. So, if you data is already loaded, it will have to reproject on the fly the entire basemap to your SR, which often causes issues.
Add the basemap first, then it should be able to on the fly project your data to it. at least, if your data really is in the defined coordinate system for that dataset.
Could be something wrong with the data then.
You say you collected with Geo7x. Did you have whatever software you are using on the Geo to capture/export the data in WGS84 UTM 13N or just the default output?
I believe Geo's normally collect in WGS84 lat/long and if not setup to output something else, that is what the output will be.
Did you do any re-projection on the data from the Geo in ArcMap?
One thing you might try is to re-define the data and assign it to WGS84 lat long (WKID: 4326), close ArcMap, open with new map, add the basemap, then add the data that has been re-defined to 4326. If it lines up, then you would need to project the data if wanted in UTM.
If you're sure the data is correct and in the right UTM spatial reference, you might export to a FGDB feature class and run the Repair Geometry GP tool. Sometimes a corrupt dataset/feature can cause issues like this.