I have not counted them - I wonder how many State Capitals do not have passenger service. I know Tallahassee in Florida does NOT have passenger service.
I count 20. Currently not served are the following:
Juneau, AK - obviously impossible
Phoenix, AZ - formerly served, now bypassed due to track abandonment
Dover, DE - Delmarva Peninsula service looks unlikely at least by Amtrak
Tallahassee, FL - served 1993-2005
Honolulu, HI - obviously impossible
Boise, ID - served 1979-1997, restoring service has a Phoenix-esque track abandonment issue
Des Moines, IA - not served since before Amtrak
Frankfort, KY - infeasible, not on any reasonable route
Baton Rouge, LA - not served since before Amtrak
Augusta, ME - not served since before Amtrak, but a plausible future extension of the Downeaster beyond Brunswick
Annapolis, MD - not on any reasonable route (Baltimore-Annapolis commuter service maybe)
Helena, MT - as discussed in this thread.
Carson City, NV - not feasible (bus from Reno to CZ has often been provided)
Concord, NH - not served since before Amtrak (anyone for Boston-Montreal service?)
Santa Fe, NM - infeasible for long distance, but the Rail Runner has been added
Bismarck, ND - lost service 1979
Columbus, OH - lost service 1979
Pierre, SD - not served since before Amtrak. The tracks are still there but sort of a dead end
Madison, WI - if it happens it'll be a northwestward expansion of Chicago commuter service
Cheyenne, WY - served intermittently by San Francisco Zephyr and later Pioneer.
I assume you'd count Olympia WA and Montpelier VT as "served" since nearby junctions are served and the main lines don't go through the capitals.