I realize from the comments that I should have said more about the specific issue of term limits for Supreme Court Justices.
The case for term limits for the lower federal judges (circuit and district judges) is weak. As I said in my original posting, the institution of "senior status" largely takes care of the senility problem. (Also, contrary to one of the commenters, incapacity is a recognized basis for removing a federal official by the impeachment process; the first federal judge impeached and removed from office was a drunkard and a lunatic, but had not engaged in wrongful conduct, such as taking a bribe.) There is shirking, chiefly in the form of excessive delegation of judicial functions to law clerks and other staff and excessive indulgence in leisure activities, such as travel, and the more serious form of misbehavior that consists of willful decision-making. However, these problems are not serious enough to warrant a fundamental change that would reduce legal certainty by increasing judicial turnover and would make a federal judicial career less attractive and so reduce the field of selection, though the latter effect could be offset by salary increases--but that, of course, would be a cost also. In addition, as I said, candidates for federal judgeships are carefully screened, and at an age when most people either have, or have not, established habits of work that will persist even if sticks and carrots are removed.
The issue of term limits for Supreme Court Justices is more challenging, first because the Court is, to a great extent, a political court, that is, a court the decisions of which are guided by the policy preferences of the judges, and second because Justices have less incentive to retire than the lower-court judges.
Let me start with the political issue. The contrast with the lower federal courts, especially the courts of appeals, should not be overdrawn. Plenty of cases that never get to the Supreme Court are political in the sense just indicated, but fewer, and they are the less politically fraught cases--otherwise the Court would take them up. The best index to the political component of a court is the amount and character of controversy regarding appointments: controversy is greatest for Supreme Court Justices, less for court of appeals judges, and least for district (trial-level) judges; and the focus of controversy is most likely to be ideology rather than politics at the Supreme Court level, less likely at the court of appeals level, and least likely at the district court level.
But I don't think the political argument for imposing terms limits at the Supreme Court level is persuasive, despite the anomaly in a democratic system of having a corps of powerful political officials who serve for life, as in a monarchy. A lesser objection is that by increasing turnover of Supreme Court Justices, term limits (depending of course on their length) would operate as a tremendous political distraction, since it would be known with certainty when a vacancy would occur. So the political struggle over a successor would start sooner, and there would be more such struggles because there would be more vacancies.
More important, when we contrast democracy with dictatorship we aren't just comparing term lengths; we are also comparing incentives. Officials who are elected for short, fixed terms and can be reelected have a strong incentive to conform their behavior to the preferences of the electorate, interest groups, public opinion, and other more or less democratic sources of influence on policy. An official appointed (not elected) for a long fixed term, and ineligible for reappointment, is a tyrant, in the sense of being (largely) insulated from the normal political constraints on official behavior.
We may want that insulation; we may want a court to be an independent power base; but the premise of the movement for judicial term limits is that courts are too independent, too powerful. The imposition of term limits would not reduce that power. It would merely increase the number of power holders. And as they would be holding power successively rather than simultaneously, there would be no competitive check on their exercise of power. So I don't see how term limits would actually limit judicial power.
But there is still the retirement question. People can get stale from serving in the same job for a great many years and most elderly people, before diagnosable senility sets in, experience a diminution in mental acuity and, especially, adaptability to novelty. The combination of very long service in the same age with very great age is likely to produce a decline in performance. And while long experience in a job can make one more efficient at it, beyond a point additional experience adds nothing.
This is certainly a problem for the Supreme Court, but perhaps not a terribly serious one. There are four reasons. First, if one's performance declines from a very high level, it may remain quite adequate; Holmes, Brandeis, Learned Hand, and other illustrious oldsters were not as sharp in their eighties as they had been, but they were sharp enough. Second, the Supreme Court's workload is very light, with a long summer recess. Third, the Justices have terrific staffs. And fourth, the most important skills in law are verbal and rhetorical, and they tend to decline with age less rapidly than logical, theoretical, and mathematical skills. On all these counts, Supreme Court judging is the quintessential geriatric profession.