Here is the best short summary of the reformed view of covenant children that I've seen.
There is no ground in the promise of the covenant either for presumption that all baptized children are regenerate (or united to Christ) or for the presumption that covenant children are reprobates until they prove otherwise. We are to treat our children as baptized persons, to whom promises have been made. We should expect our covenant children to take up Christ's benefits for themselves, sola fide. We must catechize them faithfully and encourage them to make profession of faith and to come to the Lord's Table, and if they do not, they should face ecclesiastical discipline.
Baptism and the Benefits of Christ, R. Scott Clark, published in The Confessional Presbyterian, 2006