Is there any downside to using a larger than needed smoothing capacitor?

As far as caps go, there are two competing requirements:long-term (ripple) and instantaneous (spike). A big electrolytic can give you the former but not the latter. Generally you parallel your large electrolytic with a smaller 0.1uF capable of supplying that instantaneous spike whilst the electrolytic lumbers into action. Or the 0.1uF may be for local decoupling to stabilise that regulator. If the specified capacitor is actually 0.1uF or smaller, then the intention of the capacitor is to supply small amounts of charge very fast. Do not replace this with a bigger electrolytic - that's definitely a case where larger is worse not better.

Going past that, you'll have to tell us what kind of regulators you're dealing with. If it's just a basic linear regulator then it doesn't really matter. If you have a switching regulator though, the capacitor will affect the resonant frequency of the switcher, so be very careful there.

A larger than minimum smoothing capacitor on the output of a transformer and rectifier will give you lower ripple, which is a plus. It's a small plus however, as even doubling the size of the capacitor will only (roughly) halve the ripple. Anything downstream of a large capacitor will need to have significant Power Supply Rejection Ratio (PSRR) to cope with the ripple. There are cheaper ways of improving this by a factor of two than doubling the size of the Big Filtering Capacitor (BFC).

The downside to a larger BFC is that it will draw larger, shorter current pulses from the input transformer and rectifier.

This can cause a number of problems, though most are small, or can be mitigated.

a) Higher electromagnetic interference generation, due to larger current pulses, and higher currents being switched off in the diodes.

b) Slightly hotter diodes and transformer, due to larger RMS current.

c) Poorer input power factor.

A sniff of inductance somewhere in the supply (AC input, transformer leakage inductance, post transformer or post diode) will reduce the magnitude and extend the length of the rectifier pulses, improving all of the above.

Note: my interpretation of the OPs post is we are talking about capacitors on the output of voltage regulators, some other posts seem to assume the asker is talking about capacitors on rectifiers.

The main downside of a bigger capacitor is that the switch on rise time and switch off fall time will be greater. That means more stress on the regulator during startup and in extreme cases may even cause an overcurrent shutdown of the regulator. It can also cause problems for loads which don't handle undervoltage very well.

Having said that I don't think there is any point trying to micromanage the size of such capacitors. In most cases allowing a generous margin (a factor of 2 or more) over what you think you need is unlikely to be a problem.