The map is essentially just a background image. Therefore, it is not possible to make the borders or labels bigger or darker. One option would be to turn down the transparency of the color on the color shelf. Another option would be to add the counties to your data and add them to the label shelf. This idea might be worth voting up: http://community.tableau.com/ideas/1522
Hope this helps a bit!
Is there a way to manually write on the map so I can write the names of the boroughs in bolder text so they can be seen over the shading?
Try this packaged workbook.
Note, this uses polygon mark type, but you could make it to use line type if needs be. Labels are GEOID.
As the data comes from USGS, there is also pre-joined demographic data available.