What manuscripts did people use to create the NIV and KJV? Are the manuscripts used to fashion the NIV older than those used for the KJV?

Both versions use distinct sets of manuscripts for the Old Testament and New Testament.

One might say that the NIV uses an "older" Old Testament manuscript on occasion by deferring to the Septuagint or Dead Sea Scrolls (as explained below), but I am not sure this is significant. There are rumors that the King James translators may also have done likewise, even if forbidden by the translation rules the King had put in place (see, e.g., Adam Nicholson, The Making of the King James Bible).

In the case of the New Testament, whereas the King James editors seemed to have consulted a single 1598 manuscript compilation, the NIV, which relies on the so-called "Critical Text", probably included some "older" manuscripts in its translation. But it is very difficult to judge whether the underlying text is older or newer from the age of the manuscript. A newer manuscript may actually hold a copy of a variant that had somehow been lost, for example.

With regard to archaic English language, some prefer the King James Version because it preserves a difference between singular and plural forms which has now been lost to the English language, but present in the underlying Greek.

In Mexican Spanish, for example, one uses the word "tu" for you if speaking to a single person, and "Ustedes" if one is speaking to a group. This distinction has been lost in modern English (except, perhaps, in Texas, where "y'all" is used), but it was present in Jacobian English (i.e. "thou" for you singular, "ye" for you plural. There is a discussion of this here.

I try to set out the major differences between manuscripts below.

Old Testament

NIV

The Introduction provided in the 2011 New International Version states:

For the Old Testament the standard Hebrew text, the Masoretic Text as published in the latest edition of Biblia Hebraica, has been used throughout. The Masoretic Text tradition contains marginal notations that offer variant readings. These have sometimes been followed instead of the text itself. Because such instances involve variants within the Masoretic tradition, they have not been indicated in the textual notes. In a few cases, words in the basic consonantal text [the original Hebrew used no vowels] have been divided differently than in the Masoretic Text. Such cases are usually indicated in the textual footnotes. The Dead Sea Scrolls contain biblical texts that represent an earlier stage of the transmission of the Hebrew text. They have been consulted, as have been the Samaritan Pentateuch and the ancient scribal traditions concerning deliberate textual changes. The translators also consulted the more important early versions—the Greek Septuagint, Aquila, Symmachus and Theodotion, the Latin Vulgate, the Syriac Peshitta, the Aramaic Targums, and for the Psalms, the Juxta Hebraica of Jerome. Readings from these versions, the Dead Sea Scrolls and the scribal traditions were occasionally followed where the Masoretic Text seemed doubtful and where accepted principles of textual criticism showed that one or more of these textual witnesses appeared to provide the correct reading. In rare cases, the committee has emended the Hebrew text where it appears to have become corrupted at an even earlier stage of its transmission. These departures from the Masoretic Text are also indicated in the textual footnotes. Sometimes the vowel indicators (which are later additions to the basic consonantal text) found in the Masoretic Text did not, in the judgment of the committee, represent the correct vowels for the original text. Accordingly, some words have been read with a different set of vowels. These instances are usually not indicated in the footnotes.

The NIV Old Testament omits the so-called Deuterocanonical books that were included in the King James Version, as well as in other versions based on the Latin Vulgate (i.e. Douay-Rheims) and Septuagint (e.g. the 1851 English translation by Sir L.C.L. Brenton).

KJV

The original 1611 King James Version and subsequent updates published by Oxford and Cambridge in the ensuing centuries included the Deuterocanonical books, which were written in Aramaic and Greek. I have never seen anything identifying which particular manuscripts the translators consulted for these. As far as I know, only Cambridge continues to publish a version of the King James Version with the Deuterocanonical books included.

The underlying Hebrew text is supposed to be a version of the Masoretic Text compiled by the Tunisian Jew of Spanish origin and later Christian convert Jacob ben Hayyim ben Isaac Ibn Abonijah, published by Daniel Bomberg in Venice sometime around 1525 (Introduction to the Rabbinic Bible, tr. Christian Ginsburg, p. 2-7).

New Testament

NIV

Again, according to the Introduction in the 2011 NIV:

The Greek text used in translating the New Testament is an eclectic one, based on the latest editions of the Nestle-Aland/United Bible Societies’ Greek New Testament. The committee has made its choices among the variant readings in accordance with widely accepted principles of New Testament textual criticism. Footnotes call attention to places where uncertainty remains.

The Greek New Testament the authors refer to is a compilation of hundreds of different Greek manuscripts. The editors essentially judged all of the variants available for each verse and made a decision as to which particular variant reading to select. Bruce Metzger has published separately a Textual Commentary that explains each decision made. The resulting text is sometimes referred to as the "Critical Text".

KJV

Dr. Maurice Robinson claims in the introduction to a modern edition of the 1550 Stephen's Textus Receptus that there are actually several extant Greek texts published around that time frame that are similarly named. He writes:

The Stephens 1550 edition of the so-called “Textus Receptus” (Received Text) reflects a general agreement with other early printed Greek texts also (erroneously) called by that name. These include editions such as that of Erasmus 1516, Beza 1598, and (the only one actually termed “Textus Receptus”) Elzevir 1633. Berry correctly notes that “In the main they are one and the same; and [any] of them may be referred to as the Textus Receptus” (Berry, p.ii).

All these early printed Greek New Testaments closely parallel the text of the English-language Authorized (or King James) Version of 1611, since that version was based closely upon Beza 1598, which differed little from its “Textus Receptus” predecessors. These early Greek “TR” editions generally reflect (but not completely) the “Byzantine Textform,” otherwise called the “Majority” or “Traditional” text, which predominated throughout the period of manual copying of Greek New Testament manuscripts.

Dr. Robinson also explains the key differences between the Critical Text (e.g. NIV) and the Textus Receptus (KJV):

The user should note that the Stephens 1550 TR edition does not agree with modern critical editions such as that published by the United Bible Societies or the various Nestle editions. These editions follow a predominantly “Alexandrian” Greek text, as opposed to the Byzantine Textform which generally underlies all TR editions. Note, however, that 85%+ of the text of ALL Greek New Testament editions is identical.

He also points out that the New King James Version (NKJV), published by Thomas Nelson, footnotes verses where the CT and TR variants diverge.