I always associate Government st. with ghetto. BUT that is just me because I know that North 22nd is somewhere intersected with it. Anyways. Stay away from North 22nd or areas like that ('North'-named streets all clustered together).
I'm lousy with directions and locations so don't want to alarm you but I do know that that area is not very safe at all.
If you want to know if youre close to the dangerous area (North sts), heres how to tell. One of the 'attractions' in Baton Rouge is the Casino (boat) which sits on the levy/ Rivercenter. Very nice place for entertainment, attracts tourists. Its fairly close to, if not in, the downtown area.
But if you get lost you can easily slip into the wrong neighborhood. I've been living here for 20+ years and never ventured into that area so I had no clue about the crime until I went to visit my ex on the outskirts of the area and he told me that there were shootings in the deeper area.
Not trying to scare you. There are good areas around Government st, depending on which end you're on. Just don't be near anything that sounds like North 22nd. (at least not while you're new and haven't gotten to know the area yet).
The weird thing is one of the routes eventually leads to LSU campus.
If I were moving here, I'd stick to Bluebonnet, Highland or any area closer to the Mall of Louisiana. (Do NOT confuse that with mall city or Cortana Mall). Only because I've seen less riff-raff and it doesn't have the reputation of North 22nd area.
Areas further out but very nice are Coursey Blvd or Lee Dr (that whole area is beautiful), IMO.
If $800 is your budget and you are just looking for accommodations, I would suggest looking into (but can't personally recommend because I've never been there) Bluebonnet Towers, about 15 minutes away in 'perfect' traffic.
These are closer to the Mall of Louisiana and a short commute to government depending on traffic. Because of traffic and the route you choose to take it could take up to an hour I guess but it's not very far from government st at all.
You pay one fixed monthly ($500-750 ) and all utilities are included, no lease. Supposedly catered to businesspeople but anyone can live there. Look into it if you're on a budget and don't mind a commute. I had been wanting to look into these myself but am moving out of BR asap.
Moderator cut: edit
Separate, better link:
Bluebonnet Towers in LOUISIANA - Apartments for rent | ApartmentFinder.com
Also do not get suckered into the low prices of an area called GSRI. Never lived there but told it's very dog-eat-dog so to speak, breakins, muggings and theivery sort of stuff. Not sure about murders but can't rule it out. You shouldn't have to worry about this because this is pretty far from the area you're looking for but if you happen to hear about it and $300 dollar 2 brs, then run far away.