The Chinese buses probably aren't any less unsafe than your name brand buses. Remember, the Chinatown buses are most likely operating more trips than the competition due to the low price point.
If you book well enough in advance, you can get a ticket for a buck plus fees. There are a few companies operating out of South Station Bus Terminal. Megabus, Bolt (owned by Greyhound), Peter Pan, Greyhound, Yo! (Greyhound's Chinatown bus brand), and Lucky Star (which is a former Chinatown bus brand that decided they want to serve a bus terminal instead of a random curb).
Dropoff locations in NYC vary between the bus carriers. Megabus will drop you off at a curb near Hudson Yards. Bolt, Greyhound, and Peter Pan will drop off at the Port Authority Bus Terminal. Chinese buses, Yo! and Lucky Star drop off in Flushing I think.
Bolt and Greyhound are probably your most comfortable rides, with extra legroom, wifi, and power outlets. If you travel during the off-times, you can get a ticket for a little as $10-20. Greyhound has a few red-eye buses as well.