I think they're just on the railroad bridge to scout out the water below. You can get a really good view looking down on the water. The river is tidal at that point so even if they know exactly where he entered the water, they don't know where his body might have been carried to. So if they're checking locations quite a distance from there, that's why.
(ETA: apparently they were also checking out gaps in the bridge.)
There are half a dozen bridges and walkways over the river; I think Googlemaps shows most of them though I thought there was a footbridge that isn't showing up. There are also a number of docks and boat landings where people can put in with shells, kayaks, small sailboats, etc.
https://goo.gl/maps/4xxUnhBZU6E2
The place where they've been looking, variously identified as the old Spaulding Hospital, Nashua Street, Nashua Street parking, is where you get to if you keep walking past the TD Garden/North Station building. It's primarily a commercial and recreational area, so it's pretty empty late at night and not terribly well lit.
One thing about Boston is that it's out on a peninsula. No matter where you are, if you start walking and keep walking, odds are you will end up at water. The harbor, several rivers, several channels, ponds and reservoirs, lots of stuff to drown in.
I don't think he had the intention of going to the water. No, he wasn't terribly drunk, but he was almost certainly hypothermic.
When you're hypothermic, your brain slows down. Your judgement fades. It's common for hypothermic people to latch onto one idea and pursue it through hell and high water, or keep walking with no idea where they are. Then as your body temperature continues to drop, your fine muscle control starts to go. You get clumsy. Not much after that, your major muscles stop working and you can't even walk. (
http://www.emedicinehealth.com/script/main/art.asp?articlekey=124962)
Alcohol consumption can make you lose heat more quickly.
