Why did the border states- Delaware, Maryland, Kentucky, and Missouri- where slavery was legal, not join other southern states in seceding in the months before Abraham Lincoln took office?

Because demographics do not coincide so nicely with lines and borders, there were a number of states that bucked the trend of being almost completely unionist or completely secessionist, those being the numerous border states that straddled the Mason-Dixon line.

States like Delaware, Maryland, Kentucky, Missouri, and Virginia would all have been considered border states that, depending on the geographic area, had significant amounts of both sides. So as calls for secession rang from the state houses of those in the South, these divided loyalties meant these border states had a divided government as well as a divided populace, and so instead waited to see what would happen.

After the attempted shelling of the ship Star of the West and the bombing of Fort Sumter, Lincoln made a call for 75,000 volunteers, in which case several more states joined those that had already declared secession, those being Arkansas, North Carolina, Tennessee, and Virginia, though in the latter case much to the chagrin of those in the upper area of the state. Because of the sheer amount of Unionists in the northern counties, they formed the Wheeling Convention in 1861 and broke off from Virginia, forming the new state of West Virginia, before being fully admitted in June of 1863.

The border states continued to remain divided in their loyalty and wanted to remain neutral in the fighting, but were de facto in the control of the North, especially as the Union army largely remained in control of each of the states. In the case of Maryland, after having debated to not secede but try to remain neutral by not allowing Federal troops to use their rail lines, martial law was imposed throughout the state, and a third of the legislature was arrested without charges because of fears that they would attempt to aid the south.

Those in the state also had to contend with the aftereffects of these divisions in the case of guerrilla partisan forces of both sides, such as William Quantrill's Raiders, James Lane's "Jayhawkers," Mosby's Rangers, and Charles Jennison's more indiscriminate band. Furthermore, you had proper military units raised independently for both the North and the South which fought for their respective side, several times meeting against each other, as in the case of the 1st Maryland Infantry (CS) fighting the 1st Maryland Infantry (US) at the Battle of Front Royal.

All in all, there were various factors for why these states did not secede, but the one which ties them all together was the fact that just within a single family, let alone the entire state, you had those which believed in two different ideologies, and for the most part the states sought to try and remain neutral in the conflict.