First, we must acknowledge that "Dixiecrats" is an anachronistic term in relation to your question. The States' Rights Democratic Party, better known as the Dixiecrats, was formed and dissolved in 1948. But Theodore Roosevelt was born in 1858, growing up in the Reconstruction era, and died in 1919 - thus, he would not have recognized Southern Democrats as Dixiecrats.
That out of the way, we can certainly dive into TR's views on the Jim Crow South, which were as a general rule less than favorable, although his views on race overall were more complex. But TR believed more than anything else that individual people should be treated according to what they accomplished, regardless of anything else. Did you know that even the history of the Teddy bear is tied up in TR's views on the treatment of Black people?
In 1902, TR was invited by the Governor of Mississippi to visit their state to hunt black bears. The hunt went poorly, and eventually their guide (a Black man named Holt Collier) subdued a black bear and tied it to a tree to be shot. TR refused to shoot the animal, believing the act to be unsportsmanlike (but he did have another member of the party put it out of its misery anyway).
Political cartoonist Clifford Berryman later immortalized his moment in a cartoon titled "Drawing the Line in Mississippi," and the cute little bear cub in the cartoon helped deflect some of the bad press from the hunt. (The cartoon can be seen on the TR Center digital library here. It was this cartoon that inspired the creation of plush Teddy bears.) Based on the title, and the fact that the bear bore some resemblance to caricatures of Black people, some people believed that the cartoon was also a double-entendre, playing on TR's recent crusades against lynching. Within weeks of having taken over the presidency following the assassination of William McKinley, TR was investigating a man he had appointed to a federal position, who had been accused of involvement in a lynching.
President Roosevelt remained staunch in his stance against lynching throughout his career, even though this hurt him politically in the South. In 1903, he wrote, "I should be sorry to lose the Presidency, but I should be a hundredfold more sorry to gain it by failing in every way in my power to try to put a stop to lynching and to brutality and wrong of any kind."
In addition to being fervently against lynching, Roosevelt took many other actions towards creating an environment in which Black people had more equality, at least under the law. He was, of course, a member of the Republican party, which was at this time still the party with which African Americans were aligned, thanks to Abraham Lincoln. In 1900, as Governor of New York, Roosevelt signed a legislative act for the desegregation of public schools, “no person shall be excluded from any public school in the state of New York on the account of race or color.” (This did not fully end the existence of segregated schools, it is important to note.) As president, he invited Booker T. Washington to dine at the White House, and despite the backlash from the South, never apologized or backtracked from that decision - although he never made another one. He also appointed progressive judges and (initially) encouraged the prosecution of peonage cases in the South.
Roosevelt's record on race was decidedly mixed - his discharge of an entire regiment of Black soldiers without evidence of their supposed crimes in the Brownsville Affair is perhaps the greatest stain on his presidency. His writings make quite clear that he believed strongly in social Darwinism and even eugenics. But, he also appointed Black men to civil service jobs, working in conjunction with Booker T. Washington to do so. Overall, TR believed that Black people (and all people of color) were intellectually inferior to White people, but he had no quarrels with appreciating individual Black men (like Washington) on their merits. (Please note that I do not say this to excuse his views, but rather as a factual observation.) “The only wise and honorable and Christian thing to do is to treat each Black man and each white man strictly on his merits as a man, giving him no more and no less than he shows himself worthy to have,” Roosevelt wrote of his meeting with Washington.
Roosevelt even suspended postal services in Indianola, Mississippi, when they refused to accept the appointment of Black postmistress Minnie M. Cox! He federally prosecuted those who made violent threats against her, and reduced the status of their post office from third-class to fourth-class. This, of course, was just as much about cementing his own power as it was demonstrating his views on race and racial politics. But it does demonstrate that he was not fully in favor of segregation, and that even though his efforts were uneven, he did make attempts to create a Square Deal for all people, regardless of race.