First off, make sure that you're not mistaking a cold for allergies. In early springtime especially, it can be difficult for even a doctor to distinguish the cause of a runny nose, as winter viruses may still be going around. The lone giveaway symptom of allergies, however, is itchiness of the eye, nose, and/or roof of mouth.
Children as young as 1 year of age can get seasonal allergies. It's possible that such allergies can present themselves in even younger children. But the main takeaway here is that the belief that seasonal allergies don't present themselves till around age three just isn't true. Trust your doctor if they suggest some form of allergy testing.
Allergies are passed down from parent to child, but not necessarily the same allergies. More so, it's just the susceptibility to allergies in general that is hereditary. Remember that your dust mites allergy might instead present itself as a pollen allergy in your child.
Try antihistamines that are long-lasting and non-drowsy. Examples are brands like Children's Claritin (loratadine) and Children's Allegra (fexofenadine). If your child's nasal stuffiness and drainage is especially bad, try the nasal spray Children's Nasacort.
Keep treatment consistent. Whatever your doctor's instructions, stick to them! Always giving your child his or her allergy treatment at the same recommended time of day not only ensures a routine is created, it also provides them with relief when their symptoms are worst.
Pollen-proof your child's room. Always keep the bedroom windows shut and be sure your child showers after outdoor play. HEPA filters on central AC units can also help to reduce pollen levels in your home.