The evidence is somewhat mixed. Maybe. In summary, there is conflicting evidence. None of the evidence here is particularly strong. I conclude that cat exposure is neither associated with a strong benefits nor detriments, or we might expect to see more consistent results.
None of these studies are sufficient to show causality - merely suggestive correlation.
Mild evidence that cats are good
This was only a survey but it found cat exposure was negatively correlated to allergies.
Children exposed to pets during the first year of life had a lower frequency of allergic rhinitis at 7–9 years of age and of asthma at 12–13 years. Children exposed to cat during the first year of life were less often SPT positive to cat at 12–13 years.
Evidence that cats are mildly evil
Exposure to animal dander did not influence the prevalence of clinical disorder, but positive SPT reaction to cat dander was more prevalent in infants who were exposed to cats and/or dogs (p = 0.04)
So cat exposure isn't associated with serious allergy problems, but is associated with subclinical ones.
Evidence that cats are mild
This was a study focussed on cow's milk, but as an aside:
Allergic disease was not associated with exposure to tobacco smoke, house dust mite antigen, or cats.
So cat exposure is UNcorrelated with allergies.
- Susanne Lau, Sabina Illi, Christine Sommerfeld, Bodo Niggemann, Renate Bergmann, Erika von Mutius, Ulrich Wahn, the Multicentre Allergy Study Group, Early exposure to house-dust mite and cat allergens and development of childhood asthma: a cohort study, The Lancet, Volume 356, Issue 9239, 21 October 2000, Pages 1392-1397, ISSN 0140-6736, DOI: 10.1016/S0140-6736(00)02842-7.
However, no relation between early indoor allergen exposure and the prevalence of asthma, wheeze, and bronchial hyper-responsiveness was seen.
So, cat exposure is UNcorrelated with allergies.