The delta-v's for known Earth-crossing asteroids are as low as 60 meters per second (60 m/s), as compared to the Moon's escape velocity of 2,400 m/s. There are many asteroids with required delta-v's lower than the lunar surface.
In a probable mission scenario to an asteroid, a large cargo will be launched into high Earth orbit and undergo a gravity assist by the Moon (discussed below) to pick up speed to rendezvous with the asteroid. After rendezvous of the cargo ship with the asteroid, any human presence needed would be sent by a small vehicle on a quick trajectory.
In the late 1970s, many people thought that the ideas of asteroidal materials utilization had so much merit that equipment would be developed and missions would be embarked upon by NASA. This was naive, but it was good that they proceeded with these projections, as they are exemplary. However, some of the dates of the following missions are already past.
The Amor asteroid "Anteros" (1973EC) was projected to have equipment launched to it in late 1992. Rendezvous would happen in 1993 and the equipment would be running at full steam by early 1994. After a delta-v of 1.6 km/sec, the cargo was to be enroute to the Earth-Moon system. It was to arrive in 1995 where two lunar gravity assists and a fuel thrust "capture maneuver" of 0.3 km/sec at orbit perigee would have put it into a circular orbit between the Moon and the Earth. (The 0.3 km/sec could be lowered by a third lunar encounter if so desired, but 0.3 km/sec is so small that it may be worth a little haste.)
The Amor asteroid "Eros" offered essentially the same story. The launch date was scheduled for a year later, in 1993. The delta-v would have been 1.7 km/sec and would've taken two lunar gravity assists and a capture maneuver of 0.3 km/sec.
The investigators thought that further analysis of mission opportunities and trajectories could reduce the delta-v to near 1 km/sec for the above two asteroids. On their shoestring budget, they did a limited number of calculations, and getting a trajectory under 2 km/sec initial delta-v was deemed enough to move onto other issues like analysis of the equipment needed.
In the late 1970s, a few of the newly discovered asteroids were also analyzed for rendezvous, e.g., the Apollo asteroids 1976UA (delta-v of 0.61 km/sec), 1973EC (delta-v of 1.43 km/sec) and 1977HB (delta-v of 1.06 km/sec). These calcuations were made using 1970s computers and some remarkably persistent professionals.
Since this 1970s study, using more sensitive telescopes, many more attractive targets have been found, including the asteroid 1982DB, which needs a delta-v of a mere 0.06 km/sec (i.e., 60 meters per second, or 130 miles per hour) to be captured by the Earth-Moon system.