If you are looking to find conditions, closures due to fire danger, etc., all you need is the location (ie. road number, Tnshp, Rge, Sec., stream name), and the info you are interested in. The person may or may not know the claim by name or number, as USFS does not hold the paperwork on claims, that is through BLM.
As to approach, just be yourself and introduce yourself and be pleasant with the person either on the phone or in person. You can say you are a member of the orgainization, they may know of it or not, and you can assure them you know the code of conduct on mining/prospecting, hole filling, etc. That person may have the information you want, or will put you in touch with whoever will have it. Don't expect to get the person when you call, they may be in a meeting or on vacation, out in the field, etc. If you leave a message they will get back to you as soon as they can. Most Ranger Districts will have either a minerals or recreation person, sometimes both. The National Forest main office will have both.