There are numerous ways to determine the date of a manuscript. A common indicator is the style of writing used by the copyist. There was a period of time where copying was done in all capital letters; another time when they used only small letters; and still other times when copying was done in a cursive style. A general date would be indicated by the style. The absence or degree of punctuation also indicates different time periods.
Another measurement is the writing material itself and how it was processed, whether it was animal skin such as vellum, or a plant source like papyrus rolls and codices (pages as in a book). The technology of the development of writing materials is well known and provides another indicator of age. Further, the writing material can also be submitted to carbon-14 radiometric dating to determine the age.
Finally, artifacts like coins or ceramics which are often found with the manuscripts give an indication of the time period in which the manuscript may have been copied. The methods for determining the dates of ancient manuscripts are quite well developed today and usually yield confident results.