What is the basis for the Jesus Seminar's search for the "Historical Jesus" movement?

The Jesus Seminar affirmed seven basic principles of "scholarly wisdom" in its version of the quest for the historical Jesus. It also offered several criteria for analyzing the sayings of Jesus in terms of authenticity and inauthenticity. As mentioned in the OP it also voted on the various biblical passages and sayings of Jesus and ranked them by a color code. In doing so it included the sayings of Jesus in the Gospel of Thomas, which it characterized as a fifth Gospel:

Seven basic principles of scholarly wisdom

1. Distinguishing between the historical Jesus and the stories that the gospels tell about him. ...
2. Distinguishing between the Synoptics and John, ... generally favoring the synoptics as more historical and seeing John as more spiritual.
3. Identifying Mark as the first gospel. ...
4. Identifying the hypothetical Q document, thought to be the source of material found in Matthew and Luke but not in Mark.
5. Questioning eschatological (apocalyptic) Jesus. ...
6. Distinguishing between oral and print cultures. Since Jesus lived and preached in an oral culture, scholars expect that short, memorable stories or phrases are more likely to be historical.
7. Reversing the burden of proof. ... [Rather than requiring to prove that a verse is inauthentic, evidence should be provided to support that is authentic.]

Criteria for authenticity

The Jesus Seminar treated the gospels as fallible historical artifacts, containing both authentic and inauthentic material. The Seminar fellows used several criteria for determining whether a particular saying or story is authentic, including the criteria of multiple attestation and embarrassment. Among additional criteria used were the following:

Orality: According to current estimates, the gospels weren't written until decades after Jesus' death. Parables, aphorisms, and stories were passed down orally (30 – 50 CE). The fellows judged whether a saying was a short, catchy pericope that could possibly survive intact from the speaker's death until decades later when it was first written down. If so, it's more likely to be authentic. For example, "turn the other cheek".

Irony: Based on several important narrative parables (such as the Parable of the Good Samaritan), the fellows decided that irony, reversal, and frustration of expectations were characteristic of Jesus' style. Does a pericope present opposites or impossibilities? If it does, it's more likely to be authentic. For example, "love your enemies".

Trust in God: A long discourse attested in three gospels has Jesus telling his listeners not to fret but to trust in the Father. Fellows looked for this theme in other sayings they deemed authentic. For example, "Ask – it'll be given to you".

Criteria for inauthenticity

The seminar looked for several characteristics that, in their judgment, identified a saying as inauthentic, including self-reference, leadership issues, and apocalyptic themes.

Self-reference: Does the text have Jesus referring to himself? For example, "I am the way, and I am the truth, and I am life" (John 14:1–14).

Framing Material: Are the verses used to introduce, explain, or frame other material, which might itself be authentic? For example, in Luke, the parable of the Good Samaritan was deemed to be authentic but the framing scene frame was voted down.

Community Issues: Do the verses refer to the concerns of the tge Christian community of a later date, such as instructions for missionaries or issues of church leadership? For example, Peter as "the rock" on which Jesus builds his church (Matthew 16:17–19).

Theological Agenda: Do the verses support an opinion or outlook that is unique to the particular Gospel, possibly indicating an editorial bias? For example, the prophecy of the sheep and the goats (Matthew 25:31–46) received a negative vote because the fellows saw it as representing Matthew's agenda of disdaining unworthy members of Matthew's particular community.

Color-Coding

Unfortunately, the OP's characterization of the Seminar's color-coding scheme is flippant and inaccurate. The actual version is this:

Red beads – indicated the voter believed Jesus did say the passage quoted, or something very much like the passage. (3 Points)

Pink beads – indicated the voter believed Jesus probably said something like the passage. (2 Points)

Grey beads – indicated the voter believed Jesus did not say the passage, but it contains Jesus' ideas. (1 Point)

Black beads – indicated the voter believed Jesus did not say the passage— it comes from later admirers or a different tradition.

Being personally familiar with the Jesus Seminar and its major publications, I have largely relied on the Wikipedia article for this answer. It was taken mostly from the Seminar's own sources and is a good summary in my opinion.

Further reading:

The Five Gospels: What Did Jesus Really Say? The Search for the Authentic Words of Jesus

The Jesus Seminar and Its Critics