Johnson has owned the 5ha garden since 1971 and opened it as part of the National Gardens Scheme.
From next year, visits will be restricted to by appointment only.
Johnson said: "These so-and-sos came and cased the joint, no doubt, on an open day, and calculated exactly what to do. From next year I'm afraid everyone, even burglars, will have to make an appointment."
In March thieves stole a late 17th-century astronomical sphere and urns from the site.
The bronze armillary sphere sat on a Portland stone urn base, carved with acanthus leaves and pomegranate swags and dates from about 1690. Johnson offered a £1,000 reward.
Last month thieves struck again, stealing items such as a sculpture of Flora, the Roman goddess of flowers.
Johnson is offering a £5,000 reward for the return of the artwork, in good condition. He has installed extra security and gates at the hall.
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