Who are “Security moms”?

Solution 1:

A security mom is a mom whose main concern with regard to a political candidate (and by extension, his or her political party) is which one will do the most to protect her children from terrorism.

In this case the preceeding paragraph refers to what the writer means by security mom.

...The white female vote was crucial to George W. Bush's victory in 2004, a year that was marked by the post-9/11 political emergence of the so-called security mom — a term, interestingly enough, coined by Joe Biden...

Source

Another article contains the quote by Biden:

In the post September 11 world, however, the soccer mom began to seem an anachronism. Democrat senator Joe Biden blamed his party's defeat in the 2002 midterms on its failure to listen to women. The agenda now was... "not schools, not prescription drugs. It was 'what are you doing to protect my kids against terrorists? Soccer moms are security moms now."

As for your two other questions,

Is it associated with earlier Healthcare initiative? Is the word still current?

Nothing to do with healthcare.

Yes, it is still current. Such books as Women and the White House: Gender, Popular Culture, and Presidential Politics, published in 2012, use the term.