I towed my kids in a trailer, both at the same time, but we lived in the suburbs at the time. They're adults now. My wife and I live in Manhattan now, and I've hauled cargo in the trailer a few times, and I don't enjoy it, and I don't recommend it to you unless your Brooklyn streets are quiet and roomy.
There is the kid seat that goes on the back. That is the most popular here in NYC. If you have luggage, I guess you'll need a handlebar bag or front rack with panniers or a basket on front.
You could also consider a long bike such as a Yuba Mundo or
Xtracycle, but those are well out of your budget.
Yuba Mundo:
Even nicer is a bakfiets which has a big box between the wheels. You can fit two or three or four kids in there, and boy, do they have fun.
Bakfiets:
I see bakfietsen (the plural of bakfiets) and long bikes in my neighborhood carrying kids.
Another configuration, which your kid is not yet old enough for, is attaching a banana seat over the rear wheel. I see parents taking kids appearing to be as old as ten years on this. It works well.
Some of these cargo bikes have motor options. I test rode a neighbor's pedal-assisted bakfiets. There is no throttle to control. The bike detects when you are putting in an effort, and the motor just comes on and helps. It is surprisingly comfortable. My neighbor happily rode his kids up the hills in Maplewood, NJ.
I appreciate the fact you have a budget, and it's quite reasonable, but I'm mentioning this expensive stuff in case you decide later it's worth it. Hey, you may end up with more than one kid.
Consider that you may want fenders. For many of us, they help keep us dry and clean in marginally wet weather, though they don't help much in downpours. Your kid needs a helmet, and it would be good role modeling if you wore one, too, though this being New York, I see a lot of unhelmeted parents, and I won't be calling Child Protection Services on them.
And please get yourself some excellent lights. There is a lighting and gadgets section of bikeforums.
Welcome to bikeforums and the world of bike commuting. I hope you find both fun.