Terminology note: I am going on the assumption that by 'rheumatism' you are referring to rheumatoid arthritis, a chronic destructive inflammatory condition
There is not good or sufficient evidence to suggest that 'Radon bathing' confers symptomatic benefit for rheumatoid arthritis (and possibly some other diseases)
Is radon therapy a 'helpful therapy'?
This is a tricky-ish one to answer definitively. The best I can say is that I remain (healthily) skeptical of the evidence.
A 2012 study in mice (PDF) concluded "Radon therapy ameliorates the inflammatory processes responsible for the arthritogenic activity of human TNF in mice."; however, (1) that was in mice, not humans; and (2) the numbers involved were quite small (17 vs 18) and I cannot see any analysis to suggest their results are statistically significant.
The journal Dose-Response seems to have published a number of papers on the subject:
How might radon be helpful? From the first article:
The exact mechanism of radon's effect on human body is not completely understood. However, the most favored hypothesis is that radon's action is mediated by the neuroendocrine system, stimulation of the suprarenal glands by the hypophysis, rather than by direct action on the T cells.
The last one is a general discussion of the use low-dose ionising radiation (ie not limited to radon) to treat disease. Specifically to rheumatoid arthritis, it references a 2005 review and meta-analysis which conclude there is benefit, but should be treated with some skepticism as the author is affiliated with Gasteiner Heilstollen Hospital, which offers radon spa therapy.
It seems that there may be a potential benefit for low-dose ionising radiation via neuroendocrine induction; but the evidence is (at worst) anecdotal and not extensive.
Another study from 2015 indicates that benefit of radon spa treatment may be 'due to chance':
adding radon to carbon dioxide baths did not improve pain intensity at three months but may improve overall well-being and pain at six months compared with carbon dioxide baths without radon, but this may have happened by chance.
Note: That study includes an excellent 'plain language' summary.
As far as I am aware, radon therapy is not available via the NHS in the UK.
Overall, there is no high-quality level one data to support the use of radon therapy for rheumatoid arthritis*.
Radon exposure (due to gas build-up in homes) definitely is harmful
From the National Cancer Institute:
Radon is a radioactive gas released from the normal decay of the elements uranium, thorium, and radium in rocks and soil. It is an invisible, odorless, tasteless gas that seeps up through the ground and diffuses into the air.
Radon decays quickly, giving off tiny radioactive particles. When inhaled, these radioactive particles can damage the cells that line the lung. Long-term exposure to radon can lead to lung cancer, the only cancer proven to be associated with inhaling radon. There has been a suggestion of increased risk of leukemia associated with radon exposure in adults and children; however, the evidence is not conclusive.
So long term exposure to radon causes lung cancer, and is apparently the second-leading cause of lung cancer after smoking.
As such it is important to be aware of the risks of indoor radon exposure.
So is spa radon harmful?
Given the smaller doses involved, it is hard to answer given the paucity of evidence; but I didn't find any bold, obvious warnings or studies to suggest that.
From the 2015 article on balneotherapy referrer to earlier:
we do not have precise information about side effects and complications of balneotherapy. This is particularly true for rare side effects. Side effects may include skin rash, infection and accidents, for example, slipping on wet surfaces near the bath area. The only study that reported side effects stated that they did not find any.
So I would not go so far as to label it 'safe'.
*: Credit to Graham Chiu for suggesting this qualifier.