Short answer: Write 1719
Under the Julian/Gregorian calendar system there has always been a lot of opinions on when the year starts, including having liturgical years that doesn't align with the common years. This means that in the time there is overlap it's unclear which year is meant.
Under the Julian calendar in most of Catholic Europe the first day of the year was March 25, but with the Gregorian calendar, the Catholic church moved the first day of the year to January the first (and the lithurgical start of the year to first Advent, but let's ignore that for now). Since England and several other northern European countries had broken from the Catholic church, they kept the Julian calendar for a long time. Therefore, when they wrote "January 13, 1719", the Catholic world had already switched to the year 1720. But the practice of having January 1st as new year crept into England as well.
This means that any date that falls in between January 1st and March 25 can be in one of two years, depending on if it's written with the new new year or the old new year in mind.
To solve this, they introduced double dates, to specify clearly which year it is by writing "January 13, 1719/1720", to clarify that this is 1719 old style and 1720 new style.
So, what should you write when your source says January 13, 1719? You should write 1719. The reason for this is that except with extra proof you don't know if 1718/1719 or 1719/1720 is meant. The double dates are written to specify the year exactly, and since your source didn't, then neither should you.
But what if you know by other data that 1719/1720 is meant? Well, even then I'm not sure you should use the double date format, as it's confusing. People in general tend to understand that it means "approximately", and how can something be at January 13, but only an approximate year? You only understand what double dates mean when you know about these issues. So it doesn't really clarify anything to the uninitiated, while those who know about this do not need help clarifying it.
It also hides and ignores the fact that although this is January 13, 1719 in the Julian Calendar, it is not January 13, 1720 in the Gregorian calendar. It is in fact January 24, 1720 in the Gregorian calendar.
So, unless restricted by software, which rarely supports these kind of things, I would instead mark the date as being in the Julian calendar. Depending on context you can also add more information, like a footnote. Or make a proper conversion and write both dates, if that's meaningful.
However, of course, if the source is using a double date, so should you. Preferably with a note explaining what's happening. ;-)