If Trump succeeds in winning the Republican presidential nomination—as it increasingly seems he will—it will be above all thanks to his skill in mobilizing a disproportionately white male constituency. If he is to win in November, however, he will need to broaden his appeal. First and foremost, this means winning the white female vote.

Romney, McCain and Bush did so with little difficulty. This year’s election, however, is different. Not only will Trump be running against a woman; the polls suggest that he may have already alienated a good portion of the female electorate.

To win back these women voters, it is not enough to impugn Clinton’s record [Donald Trump’s Gender-Based Attacks on Hilary Clinton Have Calculated Risk”, NYT, April 28, 2016]. Trump must also show that he will improve their lot.

Easing the child care burden on working women would be a good way of doing this. Most states already offer some form of childcare subsidy, as does the federal government (Temporary Assistance for Needy Families). These programs, however, are largely restricted to families living below the poverty line. Middle class families, meanwhile, are faced with exorbitant childcare costs for which there is little or no relief.

I, personally, would favor direct subsidies scaled to income on the European model but this would not likely play well with small government Republicans. Instead, Trump might consider using tax credits to relieve the burden on working families.

At present, working families with children in daycare benefit from the child and dependent care tax credit (CDCTC). Given the prevailing costs of daycare, however, the cap on this credit ($3000 for one child, $6000 for two or more) is absurdly low. My wife and I both work and recently had our first child, who attends daycare. The CDCTC defers our costs somewhat but it is still a struggle to make ends meet: the $3000 credit we receive when filing our taxes does little to offset the $17,000 we annually spend in daycare expenses.

My wife is sympathetic to Trump’s positions on trade and immigration but put off by what she sees as his hostility to women. But when I asked whether she would vote for him if he were to put forward a plan to reduce childcare costs, she replied without hesitation that she would. Purely anecdotal, of course, but I would be very surprised if she were the only middle class white woman who feels this way.

James Fulford writes: If I were seeking the votes of Republican women on the childcare issue, I'd offer to make both daycare and live-in nannies tax-deductible for working mothers. But even that wouldn't necessarily impress women who think Trump is sexist, anymore than Republican policies that are objectively good for blacks will make them think that Republicans aren't racist. It's a persuasion problem, and I expect Trump to solve it.

This has been covered by Scott Adams in a post called Can Trump Change the Frame on His Perceived Sexism?, Dilbert.com, April 28, 2016. Excerpt:

What do you call it when a man insults his enemies who are both male and female? Democrats call it a “woman problem.” And indeed it is. Here I remind you that reason is not part of persuasion. If the public sees Trump as having a “woman problem” then he does. So how could he fix it? Reason won’t work, but persuasion might. As a trained persuader, I’ll tell you how I would approach it if I were him. I’ll present it as an imaginary interview with Megyn Kelly. Kelly: Mr. Trump, people say you’re a sexist. Trump: What would be an example of something where you and I have different opinions on gender, Megyn? Did you catch the technique? If Trump simply lists all the things he wants to do for women, such as make the country safer and richer, that goes a little ways toward helping his case, but not enough. What he needs to do is change the frame from “This is what Trump says” to “This is what women say, and Trump agrees.” You can pick apart anything Trump says and label it sexist, the same way you can pick apart anyone who says anything. But if Trump asks a woman – especially a conservative woman – how her views are different from his, the frame changes. Context matters.

I have no idea how Trump will actually beat this "perception of sexism" (he's not actually any more sexist than most men) but I do know that it is unlikely to involve either apologizing or surrendering, as seen in the McCain and Romney campaigns.