The founders were an (intellectually) diverse group of men and it is therefore difficult to speak as to their beliefs generally about anything. Alexander Hamilton, in the Federalist No. 78., set out the reasons for lifetime judicial appointments. An independent judiciary was necessary as a safeguard against unjust or partial laws. "Periodical appointments," Hamilton noted, "would, in some way or other, be fatal to their necessary independence." He reassured the reader that there were checks on the judiciary, it would be the weakest branch (as it had neither the executive's sword nor the legislature's purse) and judges would only remain in office during good behaviour (i.e. they could be impeached).

Moreover, as I (e.g., /u/Makgraf) discussed in more detail here, the original framework for the US government involved the house of representatives elected by the people, the senate appointed by the state legislatures and a president selected by the electoral college (the method of chosing electors was selected by the states, some states appointed electors directly without an election). The general sense at the convention (and, again, it is difficult to speak of the founders as a unit) was that (after Washington) the electoral college would generally act as a body to winnow out a short list and president would be selected by the Senate which would pick amongst the list (this was changed during the convention to the house, voting by state). In that context, lifetime appointed judges is not antithetical to the vision of how the different branches of government were selected.

Democracy just means "By the people" - The justices are brought up in a way that is the result of the people. The people vote on congress, and congress decides who to pass through. Democracy doesn't require literally every federal employee has to be directly voted on. Hell, the President wasn't even supposed to be directly voted on. The EC was intended to be voted on, as in, we vote on an educated expert to vote on our behalf. It didn't last long that way, but it's intent was POTUS wasn't even supposed to be directly voted for