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Ultimately, what fields are printed strongly depends on your citationstyle.
The logic in the style decides which data is used to be printed. Apparently your style dictates to prefer date over year, which is fine, because you would not want to have something like “1987, 1987-1998” in your references… -
The old BibTeX standard (no active development) uses the “year”, “month” “day” fields, whereas the newer BibLaTeX standard (active development here) introduced the “date” field, which is supposed to follow ISO 8601 syntax (e.g now supports date ranges and time). Of course, if you still have data in the “year” field, you can use that, but if push comes to shove, the “date” field might be the way to go, especially if your citationstyle prioritizes the “date” field.
JabRef offers the possibility to move data from selected fields to other fields, e.g. from year field to the date field. Check out edit > automatic field editor. (Make sure to have a backup of your library, before making huge changes!)