Introduction Anxiety in women is highly prevalent during pregnancy and in the postnatal period. Anxiety disorders in mothers have been linked to adverse outcomes in their children’s development. However, large-scale prospective studies on this issue, covering both the prenatal and postnatal period with follow-up periods beyond the first year of life are scarce. Method In this prospective cohort study, data gathered from 1336 Norwegian women and their children were used. Maternal anxiety symptoms were measured at gestation week 17–19 and 32, as well as 8 weeks postpartum using the Symptom Check List. Child development problems were assessed at 2 years postpartum using the Ages & Stages Questionnaire: Social-Emotional. Logistic regressions were conducted to examine the association between maternal prenatal, postnatal, and perinatal anxiety and the risk of social-emotional development problems in 2-year-old children. Results Of all women, 8.2% experienced prenatal anxiety, 4.0% had postnatal anxiety, and 4.4% reported perinatal anxiety (i.e., anxiety in both the prenatal and postnatal period). 5.6% of the 2-year-olds showed problems in their social-emotional behavior. Child development problems were associated with maternal prenatal anxiety (Odds Ratio [OR] = 2.48, 95% CI 1.55–4.92), postnatal anxiety (OR 3.32, 95% CI 1.43–7.74), and anxiety both in the prenatal and postnatal period (OR 3.98, 95% CI 1.85–8.56). Adjusted for confounders, maternal anxiety continued to be a significant predictor of adverse child social-emotional development (postnatal anxiety: OR 2.46, 95% CI 1.01–5.97; perinatal anxiety: OR 2.40, 95% CI 1.03–5.59). Discussion Maternal postnatal anxiety and anxiety both during and after pregnancy are unique substantial predictors for problems in a 2-year-old’s social-emotional development, even when controlled for confounders.