Trump also advocates eliminating the federal minimum wage, leaving it to states to set the rate, and repeal the Affordable Care Act.
"With somebody like Donald Trump you would see a race to the bottom in this country with working families paying the price," Clinton said.
She said she wants to "reject that vision and come up with a much more positive one for families and children."
During her tour of the health clinic Clinton was told that the Affordable Care Act had greatly increased the percentage of patients with health coverage.
"If it ain't broke, don't mess with it," Clinton said.
Kentucky holds its Democratic primary on May 17. While polling is scant, rival Sen. Bernie Sanders is favored in the state. Losses this month, including an expected defeat in the West Virginia primary on Tuesday, would slightly erode Clinton's lead over Sanders but not change the overall dynamic that all but ensures she will be the nominee.
Clinton's focus on family services and the cost of child care may have less to do with Sanders and more to do with taking a policy-heavy high road against likely Republican nominee Trump.
Earlier in Lexington, Clinton visited a city social services center that offers subsidized day care for infants and young children.
It just doesn't make sense," Clinton said there, referring to the cost of quality care. "It’s the most important job that any of us can do, and we’re making it really hard and really expensive.”
The Democratic presidential front-runner's campaign said the proposed mix of federal subsidies or tax credits to pay for the new benefit will be announced later. Clinton also proposed raising wages for child-care workers and expanding home-visit programs for new parents.
Two-earner families in Kentucky now pay about 20 percent of their income for child care, she said.
Her campaign said that while middle class incomes have stagnated over the last decade, the cost of child care has risen nearly 25 percent.
Clinton wants "substantial new investments" in federal subsidies for lower-income families and tax breaks for middle-class families to meet the goal of limiting costs to 10 percent of family income, campaign policy adviser Ann O'Leary said.
Clinton said the expanded benefit would be offered on "a sliding scale" based on need, but did not provide an estimated cost.
"I'm looking for good ideas" that can serve as national models, she said during a discussion with working parents at the Lexington center.
In brief remarks to reporters there, Clinton said existing child care subsidy programs like the one in Kentucky and another at the federal level that could be expanded.
"We already have a system. It's just not kept up with the times. It hasn't kept up with the cost," Clinton said.
She rued the tradeoff many parents make - choosing to pay for quality child care for very young children but putting off saving for college.
"You've got to look at what families are facing today. And I think, still, too many of the programs are really designed still for an earlier time that just doesn't exist anymore," Clinton said. "We don't have that kind of family structure," she said, referring to the formerly traditional model of a stay-at-home mother.
Now, families juggle not only to pay for expensive care but how to handle gaps in that care, such as snow days, illnesses and summer vacation.
"It's time to face up to the reality of what family life is like today and to support families," Clinton said.
On Monday, she deflected reporters' questions about whether it is fair or appropriate for Trump to raise the issue of former president Bill Clinton's infidelity.
“I’m going to let him run his campaign however he chooses," she said in Virginia. "I’m going to run my campaign, which is about a positive vision for our country with specific plans that I think will help us solve problems that we’re facing."
Also Tuesday, Clinton proposed a program to help fund higher wages for child-care workers, whose wages often remain low despite the overall high cost of child care in institutional settings. The program, like one she has proposed for caregivers for the elderly and disabled, would fund and reward states for helping raise caregiver wages.
She also proposed expanding home-visit services for more than 2 million parents and young children over the next 10 years to improve maternal and child health and children's development and learning. Her campaign said Clinton wants to double federal investment in "evidence-based" home-visit programs in which social workers or nurses provide instruction, tips and resources.