To understand what a cross breed may be like you need to look at the good and bad traits of both breeds and understand you may get the best of both, the worst of both, or a mix.
Pugs: usually pretty calm and social, won't enjoy fast paced or boisterous games, eyes are a weak point and can be vulnerable to harm if children play rough or inappropriately with the dog. Any dog when hurt can bite so children need to know how to be gentle and calm with the dog.
Beagle: Also normally good natured IF you get the right lines - there are some lines with behavioural problems, tend to be good with children, boisterous, destructive (can chew children's toys etc), energetic.
All dogs are individuals so not every pug is like above etc.
Worst case scenario is a grumpy, boisterous dog with vulnerable eyes.
Best case is a fit, energetic dog who can play with children but remain good natured.
Or something like that...