Nobody knows why, but fresh strawberries are a common allergen, especially in babies under 6 months of age. Some suspect that the protein responsible for turning strawberries red is the culprit, but whatever it is, many babies develop a rash around the mouth and face after consuming fresh strawberries. Strawberries can also contribute to nappy rash although this is down to acidity rather than allergens.
Cooked strawberries very rarely cause allergic reactions so you will frequently find strawberry deserts and purees for the youngest children. These are usually perfectly safe as the cooking (and often pasteurising) process destroys whatever causes reactions in the first place.
If you wish to introduce strawberries into the diet of your young children then be aware of their allergic nature and observe your children after feeding strawberries to look for adverse signs. If your children react then leave it for a few months and then try again. The majority of children grow out of any allergic reactions by 6 months and very few display adverse signs after 12 months.
White strawberries and the 'pineberry', which some supermarkets have introduced over the last few years, do not appear to cause allergic reactions, perhaps strengthening the argument that it is the red pigment in strawberries that is to blame.