Primarily, the decision to charge for certain forms of entertainment yet offer others without cost to the visitor requires a degree of subjective judgement about their worth. For some, a museum is far more important than a trip to the cinema, yet it could be argued that they are equally deserving forms of artistic entertainment.
In addition, there are far more deserving areas that are in desperate need of funding, such as healthcare. The cultural importance of looking at art simply does not equate with the importance of ensuring that citizens are healthy. In many countries, a visit to a museum is free, yet the cost of a university education is prohibitive.
Yet these points must be balanced with the understanding that unlike other forms of entertainment, the number of people visiting museums who would be wiling to pay a significant entry fee is low. Given the choice of a museum trip or surfing the internet, the majority of people in recent generations would likely opt for the latter, meaning that the galleries would be forced to close if they were not funded.
To conclude, although there would be clear benefits to charging a fee for entry to such artistic venues, it is unlikely that one would be levied as this would almost certainly result in their closure. The best course of action at this point would be entry by donation, rather than a fixed fee.