You just need to buy an inclusive day travelcard from Basingstoke. Also known as an “outboundary travelcard”,
If you buy from the
GWR▸ website, it will be shown in the ticket types as London travelcard zones 1-6. Prices vary according to date and time of travel. There's a Sunday Out travelcard for £27. Weekdays seems to range from £31.90 after morning peak to £61 during morning peak.
That will include buses and rail/tube/DLR▸ in zones 1-6. Sticking with paper is better than using the Oyster▸ or contactless alternatives as they can be confused if you spend all day travelling around and accidentally exceed certain journey time limits.
(It’s a bit more complicated if you want to go outside zone 6 though…)
I'm not sure the journey time limits apply for travelcards. I thought that was just for Pay As You Go. Paul is correct in saying that you won't be able to load a ticket to Basingstoke on an Oyster card nor use contactless to that station. If you have a GWR or
SWR» touch card you may be able to load the combined Basingstoke and travelcard to that. The ITSO symbol is showing next to that ticket type.