The airbase is under USAF military jurisdiction. Basically the USAF major will ask himself "Is your request really necessary for the continued military efficiency of the base?" No, you will not be granted permission to enter ASI as a private pilot.
The island may be British but the UK/US have a treaty whereby the US who built the base are exempt from all charges and leases.
Feel free to contact Patrick AFB Florida.
For access to St Helena see the ATNS website [under ATM] and "HOO" at the airport. There is NO avgas, oil or spares.