This is my opinion so it may not be entirely accurate:
The Gardner Museum was run by old line Bostonians and their money. The problem was their money was tied up in trust funds and they probably weren't getting much cash from the funds anyway. The board was moribund and cheap, hence the museum was a mess, no climate control, bad security system. They could only afford to hirer college kinds who got high there, leading to the break in. The breakin was the catalyst for a revived museum. People with money and drive were invited to join the board and I would bet some of the dead weight was removed. I think a lot of the old board were descendants of descendants of people picked by Mrs Gardner herself.
My guess is that the same thing has put the Horticultural society in the same position. If the leadership is not there then it may go out of business or become a much less ambitious organization. I expect that they will find a way to recover. Didn't both the Aquarium and Zoo both recently loose their accreditations? On the other hand the Computer Museum and the Immigration Museum have both closed. Only the Horticultural Society can be blamed for their situation.