I would suggest Fern Gardenia, an excellent budget resort near Palolem beach. Palolem is a safe and clean beach, Fern Gardenia resort is one of the best at Palolem. Resort bills itself as a nature resort with plenty of greenery around and even rocks in some cottages. Kids will love splashing in the kiddies pool, older kids will love swimming and surfing at the safe Palolem beach.
Though, the resort is not on the beach, access to Palolem beach is never an issue thanks to free shuttles to the beach.
If budget is not an issue, stay at Park Hyatt. Their Camp Hyatt, meant for children is a big hit. Huge well maintained lawn, kid pool, access to one of the best secluded beaches in South Goa - Arossim makes Park Hyatt the best choice for those traveling to Goa with Children.
Another equally good option in Northern Goa is Taj Holiday Village at Candolim. Taj Holiday Village has dedicated play area and other children favorites like adventure trails and air-hockey. Both Park Hyatt and Holiday Village offer baby sitting facilities.