I am by no means an expert on Roman domestic architecture, but I have access to the right resources, so I can provide a quick answer on what my brief-ish scholarship overview says (though if someone who has really dwelled on this topic can provide a more educated answer, great!)...

So impluvia (plural of impluvium) we know best through houses preserved in Pompeii, although they certainly were incorporated into Roman houses elsewhere, too. Impluvia actually aren't just decorative elements - although those who could often embellished them with lavish mosaics etc. - but part of a very conscious water management system. The water never stood still for very long in an impluvium and it was meant to be very shallow, so mosquitoes and algae weren't really a problem in well-designed impluvia.

Pompeii as our example case is situated in an area that gets very heavy seasonal rainfall during some months, but then hardly any in esp. the August-September, late Summer season. So, when designing a Pompeian house, you need to incorporate solutions for directing rain water somewhere where it doesn't do any damage to the house structures, and if you can direct it somewhere where it can actually be used and stored, great. The groundwater in Pompeii is as deep as 20 to 30 meters underground, and not of very good quality under all those lava banks and pumice layers. Digging wells was therefore too expensive, too difficult and too pointless for everyday domestic usage. So, Pompeian households relied heavily on rainwater, and impluvia were part of a larger system for collecting and storing rainwater as effectively and passively as possible. You find them in Pompeian atria (plural of atrium), that also had a sloped-roof opening to collect water into the impluvium, known as compluvium. [The etymology of these words actually gives away the purpose of these structures: compluvium comes from con- + pluit, "it rains/flows together", i.e. it collects rain flows into one place; and in + pluit, "it rains upon"]. Can't seem to find a good photo online of a Roman house that illustrates this, but this 19th century washing house from Bourgogne, France has basically copied the exact Roman design. The fancier compluvia had a small border 'fence pool' (?) fitted with water sprouts in the shapes of various animals and beasts, like in this fine compluvium-specimen from Casa Sannitica in Herculaneum.

After getting the rain water from the roofs to the impluvium, the water still needed to be moved somewhere to get filtered and stored. Pompeii actually must have hundreds of water cisterns underground, but only a couple of the water management systems have been excavated. What we do find in the houses that have been dug down to these layers, is that impluvia usually had two different drains. The first one led directly to the street, and this would be manually opened after the first rain fall: esp. after long periods of draught, the first water coming via dusty roofs was too dirty to use, so the first batches of water could just be discarded to the streets. Once the water was clear, the second drain was opened, which lead to large subterranean water cistern under the domus designed for longer-term water storage. Sometimes, instead of a larger second drain to the cistern, you can find that the impluvium pavement is made of porous material, or with conscious cracks and e.g. layers of sand, which filter and purify the water as it slowly drips down to the underwater cistern. These cisterns were lined with water-proof plaster and kept the water cool and clean; and, somewhere in the house there would be a hole with a removable stone cover known as puteal, where one could collect water for everyday use by lowering buckets down to the cistern.

So in summary: there was not much fear of mosquitoes and algae, because the water from implivium was very consciously and effectively cleaned and collected to be stored to underground water storages - rain water was never meant to stand still for very long in the atrium!

Sources:

Keenan-Jones, D. (2005). "Water supply and wastewater disposal in Pompeii: an overview", in Ancient History: Resources for Teachers, 34, 149-158.
Jansen, G. (2017). "Water and water technology in Roman Gardens", in (eds.) W. Jashemski et al., Gardens of the Roman Empire. Cambridge University Press.
Jansen, G. (2007). "The water system: supply and drainage", in (eds.) P. Foss and J. Dobbins, The World of Pompeii. Routledge.