Source: from study

Age-Related Attenuation of Dominant Hand Superiority, PLoS One, December 2006

The clear right-hand advantage present at younger ages changed to a more balanced performance in advanced age.
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In contrast to younger subjects, the elderly employ both hands with equal frequency.
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Most elderly subjects are unaware of the changes in hand dominance that occur over their lifespan, i.e., a shift to ambidexterity.

Why? Yes, people’s brains age, but not more one side than the other (as was found). Here they suppose it has to do with practice:

If neurophysiological evidence does not support asymmetric aging of the hemispheres [which they confirmed in primate studies], a possible explanation for the move towards ambidexterity in upper limb movements with age comes from the concept of use-dependent plasticity. The advantage of the dominant hand is determined early in life, and is intensified by practice through everyday activities. When these activities decrease after retirement, or by the limitations in older age and sedentary lifestyles [16], [60], [61], it is conceivable that the practice-based superior performance of the right hand is no longer maintained, thus approaching the performance level of the left hand. This is in line with the more balanced use of both hands in everyday-life of aged subjects we found.