Family Support Centers Act of 1988 - Directs the Secretary of Health and Human Services to make grants, on a competitive basis, to eligible organizations to improve the quality and availability of health, family, and preschool services for families with one or more children under age 13.

Sets forth eligibility requirements for organizations. Requires that such organizations be located in or in a reasonable proximity to a high risk area, i.e. an area with a significantly high rate of: (1) infant mortality; (2) teenage pregnancy; or (3) high school dropouts.

Requires organizations receiving grants to offer at least five of the following services at locations convenient to the local community (for example, in a housing project, a community center, or a building occupied by a religious organization): (1) prenatal and postnatal care; (2) nutritional lunches; (3) on-site nurse practitioner services; (4) family drop-in center; (5) Head Start program; (6) parenting education services;(7) drug abuse prevention services; (8) family-based child care services; (9) center-based child care services; (10) after-school care for children under age 13; and (11) other services which the Secretary shall specify in regulations. Directs the Secretary to give primary consideration, in selecting grantees, to the location or locations at which the organization provides services to the local community.

Sets limitations on: (1) the total monetary amount of all such grants; (2) the maximum percentage of project start-up costs which may be covered; and (3) the maximum amount per grantee.

Authorizes appropriations for such grants for FY 1989.