Should I babysit a child who has recently been diagnosed with pneumonia, or will I catch it?

You can baby sit. In general, pneumonia due to bacteria is not contagious. If the child has a viral infection or influenza (the "flu") however, this could be contagious. If the child has been diagnosed to have a bacterial pneumonia, then it would be safe to babysit the child. The only major type of bacterial pneumonia that is contagious is tuberculosis, but in general, people do not refer to this as pneumonia.
Hand washing. If the child is not coughing, then wiping the surfaces in the household with an anti bacterial wipe and using a hand sanitizer will help prevent illness. If there is a cough, airborne droplets may be inhaled and cause illness. .