Enterprise NX-01 is the first deep space exploration vessel, since Warp 5 vastly increases the number of worlds Starfleet come into contact with. Travelling hundreds of light years in as much time as it took on a cargo round trip run between a handful of systems.
Before that, they'd still have ships out patroling back and forth between human colonies, just dozens of light years in every direction away from Earth.
Archer talks about having been an XO and disagreeing with a superior officer's command style, which Reed would be more used to. "Minefield" I think. So it's not like they're the first in some respects. Just that their mission is more open-ended, more expanding horizons and less about being routine traffic.
Between 2050 and during Enterprise's timeframe - 2150, there must be other pioneers and explorers setting up home away from home. Like Proxima Colony (that would be our nearest neighbouring system right?) and Vega Colony. That's probably where the gritty space western was happening for a time, until Starfleet's Cavalry brought law and order. Then there are low warp Earth Cargo Ships having to negotiate disputed terrority further afield and still facing occasional acts of piracy, while trading with Vulcans and other worlds within relatively close range of us.