Provide the tools attorneys need and use, and provide the training to use those tools. Once the attorneys are hooked on the computer, they will ask for more as they learn. But, without a doubt, the first thing is to determine if a problem exists in the firm. Some firms have a technology committee. These folks are your first line of defense.
Successful firms have one or more partners on the technology committee.
This is where your firm's management commitment comes into play. The second thing is to realize that not every computer illiterate attorney can be dealt with in the same fashion. The earlier examples provide you with some common excuses. The third thing to realize is that not everyone you approach in the office will be ready at the time you approach them. Have a contingency plan, and let the plan fall into place.