Following is based on a summary of what I recall from Ramba"m hilchot Birkat Hachodesh. I will, iy"h, add more specific info, later.
Ramba"m mentions that the primary concern is to have Pesach (15 Nissan) occur in the spring (i.e., after the equinox) since the Torah refers to Pesach as Chodesh Ha'aviv. The concern is not regarding the 1st day of Nissan, so, in most cases, that date occurred before the equinox. Sanhedrin would declare a leap year if the 15th of Nissan occurred before the equinox, Since this adds an extra month, by doing this, Rosh Hodesh Nissan now most likely occurs after the equinox, as well.
Sanhedrin could also declare a leap year for other reasons such as the barley has not yet ripened. Barley was needed for the omer offering on the 16th of Nissan. If heavy rains knocked out bridges so that it became too difficult for many people to make the pilgrimage to Jerusalem, Sanhedrin may declare that year a leap year, too even if without it, 15 Nissan would be after the equinox, anyway.
Currently, we are on a fixed calendar. In most non-leap years, Rosh Hodesh Nissan occurs before the equinox. When Pesach is at its earliest (I think the earliest Gregorian date I have seen is March 26) it is barely after the equinox. Of course, if it is not, that year, on the fixed calendar becomes a leap year.
If you look at the Judaic fixed calendar historically, you will notice a "seasonal shift" in that the 1st day of Pesach is slowly getting later on the Gregorian calendar. (There's a separate M.Y. question discussing this, if you're interested.) Thus, currently year 8 in the 19 year Judaic calendar cycle has the latest Gregorian date for Pesach. This year is a leap year, which, currently, could be eliminated as without it, Pesach would still be after the equinox. However, when the fixed calendar was created millennia ago, that year needed to be a leap year. We haven't changed the fixed calendar system at all. However, assuming that we kept it this way eternally, we would need to adjust things as at some point Pesach would occur in the fall season.