Reddit mentions of Dahon Vybe D7 Folding Bike Obsidian with Fenders

There are a lot of folding bikes under $800. You can even get electric ones for $800 and possibly less https://ridegenesis.com/collections/electric-bicycles/products/commuter-electric-bicycle-white?gclid=CjwKCAjwuvjNBRBPEiwApYq0zhHID1U_tRwm9SpqQUPCKSKAu3ViV0J5cfy4Rv_v4tF5pB79UXdPVxoCW_UQAvD_BwE

Most of Dahon's line up is under $800 as well
https://www.amazon.com/Dahon-Vybe-Folding-Obsidian-Fenders/dp/B00DYX1XOW/

https://www.amazon.com/s/ref=lp_2588037011_nr_scat_3405101_ln?srs=2588037011&rh=n%3A3405101&ie=UTF8&qid=1505636199&scn=3405101&h=c5286098a6ef1d1202c9379646b71b1f55c0d813

It shouldn't be hard to find a sub $400 bike to commute that distance on. My preference is for steel-frame bikes because they give a great comfy ride. There are many styles of bike that will serve your purpose.

In fact I did the same distance commute on a low-end mountain bike and had a great time even though it was hardly the ideal bike for commuting. When I bought that bike, my budget was also $400. Looking back, I wish I'd spent more and gotten a better bike because the commute made me fall in love with bikes again and I went out and spent twice that a little later. My commute was on a bike trail and there were few stops and starts. I'm guessing you'll have a lot of stop and start. You might need something geared lower than a road bike for that.

In traffic, I prefer a flatbar because my hand is right there by the brakes if needed and I like the greater stabiility. But I'm sure there are plenty of drop bar riders who have no trouble in the meanest traffic.

What kind of bike did you ride as a kid? You may be more comfortable on a style that you're familiar with, all other things being equal. What accessories/gear/clothing will you need (rack, water bottle cages, repair kit, visibility, etc.)? Do you plan on maintaining the bike or taking it to a shop? do you like to ride as fast as you can or do you prefer a more leisurely pace?

Just to get you started, here are some different bikes that might fit your needs. (I was impressed with the number of good bikes on craigslist in your area):

City Bikes

https://www.retrospecbicycles.com/products/kinney-flat?variant=9374985797

https://cnj.craigslist.org/bik/d/60cm-giant-rincon/6306319698.html

https://www.linusbike.com/products/rambler-7

http://breezerbikes.com/bikes/details/liberty-r-1-3-plus (way over $400)


Single Speed

https://www.brooklynbicycleco.com/collections/commuter-bikes/products/bedford-single (My fav bike to ride in city traffic is my single speed).

https://www.raleighusa.com/rush-hour-4973


Road Bikes

https://newyork.craigslist.org/stn/bik/d/new-like-miyata-classic-road/6306866055.html The narrow tires don't provide the stability offered by other bikes in this list.

https://newyork.craigslist.org/brk/bik/d/specialized-sequoia-touring/6287946883.html

MTB/Flatbar

http://civiacycles.com/bikes/venue

https://newyork.craigslist.org/brk/bik/d/giant-escape-3-bicycle/6302209792.html

https://newyork.craigslist.org/que/bik/d/specialized-sirrus-road-bike/6291789047.html


Folding

https://www.amazon.com/Dahon-Vybe-Folding-Obsidian-Fenders/dp/B00DYX1XOW. These are popular with people who have small apartments or have to carry the bike up a lot of stairs.

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(the list got out of hand, sorry. you would need to make sure any used bike is in sound working condition and fits you and it doesn't hurt to look at a few reviews on any bike you consider.)