If you are going to leave it to your heirs or to chance, compile some information on collectors or dealers who would be interested in taking some, or all, of your collection. Let your family know which records you have that are particularly valuable and how much they might expect to get for your collection. You should also keep in mind that selling a collection, particularly if you want to get full value for the collection, is a great deal of work. Unless you get an institution, collector or dealer who is willing to take it all, keeping in mind that dealers have to make a fair profit on what they buy, then you will have to make a list, find a way to circulate it, and then organize the sale and send out the records to the buyers. It takes a lot of time and there aren't many people willing to undertake such task.
The easiest way to dispose of a collection is all-at-once. That is, if you can find a buyer or institution that will take the whole thing, it is easiest on you. Unfortunately, price suffers. Piece-meal is the other option, but it takes much longer and doesn't guarantee that you'll be rid of every record anytime soon. If you have a large, high-quality collection, you might want to contact Kurt Nauck. He is very respectable and will handle everything to everyone's satisfaction. If you want to sell it yourself and you have a lot of high-quality stuff (>$5 per record), you can list them in VJM or Record Finder (or even on the Net). Sometimes even lesser-quality records can be listed in these publications or on the Net.
Donating is also an option. Universities, libraries, archives, and private public collections are all alternatives. If you think you want to donate to an institution, check it out and find one that is willing to take your collection now or at some time in the future. A collection doesn't have to be donated intact to one institution. It can be split among several. The Center For the Study of Popular Music at Middle Tennessee State University (Murfreesboro, Tennessee), Univerity of Texas Fine Arts Library (contact Dr. Carl Miller,) Library Of Congress, National Library of Canada, Rutgers, Yale, Goodspeed Opera House, Cornell, and Stanford are only a few of probably many that accept collections. Some require that collections be cataloged and computerized before they will accept them. Keep in mind that most charities are well under-funded. Retrieving information from them can be difficult or impossible for quite some time (or indefinitely) after the initial donation. Some can also be unwilling to participate in reissue projects for various reasons. Be sure to match your collecting interests with an institution's strengths, realizing that those that do accept 78s may have extensive collections already. Really, few libraries accept records other than CDs. Most librarians and archivists are conscientious enough to tell you up front when your material might be queued up, inappropriate, or unwanted. They should be able to show potential donors how collections have been handled in the past and what access and copying facilities are available and what their policies are.
When making a donation of materials to an institution, you might consider donating the funds to have your collection cataloged. Donate the $$ to get staff hired, however ephemerally. Sometimes, even a small monetary donation along with a collection can do a lot to defray the costs of the institution acquiring it.
Institutions may or may not be the best place for your collection, depending on what you want done with it. If you are looking for long-term, stable conditions from which future generations might benefit, libraries and archives are uniquely suited to that. A good number of collectors feel that giving it to an archive and putting it in the garbage are nearly equivalent in terms of the number of people who ever hear the records again.
There is no perfect or easy way to find a home for record collections, unfortunately. For more information, consult articles in the ARSC Journal 1995 Vol. 26 No. 1 pp 53-61, the Antique Phonograph News January-February 1996, or the IAJRC 86 page monograph ("Assessing, insuring and disposing of jazz record collections", $10.95, Order from Charles H. Waters, Jr., 8809 Echo Valley, Houston, TX 77055)