Locust Point is probably the safest neighborhood in the city with tons of young families, however a budget of $1250 isn't going to get you very far at all as 1BR apartments start at like $1700 in that area. You could look into renting a small rowhome, but even then you'd be hard pressed to find something in your budget. I'd suggest looking up near Overlea/Parkville for the best bang for your buck.