Did ancient people regularly suffer from colds?

Short Answer

It is difficult to determine how common or regular colds and especially influenza were in ancient times, primarily because the most obvious symptoms can be identified with a large number of diseases. Also, descriptions in ancient sources are often unclear. However, there is some indication from ancient Chinese and Roman sources in particular that colds (of which there are many kinds) were not uncommon, and it is not impossible that these descriptions included cases of influenza. Concrete proof of influenza in ancient times, though, is lacking. Also, there is little evidence of epidemics for either colds or influenza.

Details

A good example of the problem in determining what exactly ancient authors were talking about is Herodotus' Cough of Perinthus, referred to in the Wikipedia link in sempaiscuba's comment above. Compare Wikipedia's citation with the following scholarly observations:

Historians of medicine and philologists have over the years attributed the Cough of Perinthus to diphtheria, influenza, epidemic encephalitis, dengue fever, acute poliomyelitis, and many other diseases.

Source: Paul M.V. Martincorresponding and Estelle Martin-Granel, '2,500-year Evolution of the Term Epidemic' (Emerging Infectious Diseases, 2006 Jun; 12(6): 976–980)

This lack of scholarly consensus is also noted in the article Insights into infectious disease in the era of Hippocrates. (2008), and again here:

In 412 BC, in the "Book of Epidemics", Hippocrates described a putative influenza-like illness syndrome called "fever of Perinthus" or "cough of Perinthus". While some scholars claim that this is probably the first historical description of influenza (a winter and a spring epidemic of an upper respiratory tract infection occurring regularly every year at Perinthus, a port-town in Marmaraereglisi, a northern part of Greece, now Turkey), others, including the notable 19th-century editor of Hippocrates, Émile Littré (1801-1881), think that a diagnosis of diphtheria would better fit the description of complications

Source: Barberis et al., 'History and evolution of influenza control through vaccination: from the first monovalent vaccine to universal vaccines' in J Prev Med Hyg. 2016 Sep; 57(3): E115–E120

The article Historical thoughts on influenza viral ecosystems notes that the problem of description is widespread:

The classical literatures of Greece, Rome, and other ancient civilizations, as well as records from the Dark and Early Middles Ages, are packed with reports of human and animal disease outbreaks. In most cases, the diseases in question are so vaguely described that identification with modern diseases is difficult or impossible.

Source: David M. Morens and Jeffery K. Taubenberger, 'Historical thoughts on influenza viral ecosystems, or behold a pale horse, dead dogs, failing fowl, and sick swine', in 'Influenza Other Respir Viruses.' 2010 Nov; 4(6): 327–337

Kyle Harper (a classicist), in The Fate of Rome: Climate, Disease and the End Roman Empire, is more certain that influenza existed in ancient Rome but calls the lack of evidence "puzzling". In short,

Although influenza could be among the older diseases of civilization, acquired from pigs or ducks or other animals thousands of years ago, there is no clear evidence of its spread among humans until Europe's Middle Ages, and no undeniable evidence until the fifteenth and sixteenth centuries.

Source: Kenneth F. Kiple (ed) 'The Cambridge World History of Human Disease'

On colds, there seems to be a consensus, albeit tentative, that these has been around for a very long time:

Nothing certain is known about colds in the Stone Age.

Despite our lack of specific knowledge, we can be relatively sure that colds existed at that time. Once humans started living in large families and village-like communities, the ground was laid for the cold to spread, because where people live close together, colds thrive.

Source: Isabel Atzl and Roland Helms, 'A short history of the common cold'. In Ronald Eccles and Olaf Weber (eds), 'Common Cold' (2009)

The same authors note that,

For millennia, traditional Chinese medicine has believed that a cold is an illness of ‘wind and cold’...

In addition to ‘wind-cold’ with symptoms like severe chills, low fever, neck pain, no sweating (dry skin), ‘wind-heat’ (few chills, more fever, noticeable sore throat, slightly moist skin, pulsing headache) as well as various influences of moisture and dryness can also be diagnosed. Chinese medicine therefore does not treat every cold the same way.

Observations on the prevalence of diseases according to season was observed during the Han dynasty (206 BC–220 AD):

Disease in the Han dynasty became above all a seasonal phenomenon. Earlier observers had already recognized that different afflictions tended to characterize different seasons. The Zhou li (Rituals of the Zhou dynasty) observed, for instance, that headaches were prevalent in spring, whereas summer was characterized by the spread of scabieslike itching, autumn by malarial and other fevers, and winter by respiratory disorders (Lu and Needham 1967). Also widely noted was the fact that the same climatic pathogens of wind and cold had different effects in different seasons. Unseasonal weather, such as cold in summer, posed special dangers and often engendered epidemics.

Source: Kiple (ed)

Evidence also comes from, for example, the Chinese physician Zhang Ji of the 2nd century AD.

As for cold, the study of the feverish disorders to which it gave rise would eventually become the subject of the most influential treatise of the pharmacological tradition, Zhang Ji's Shanghan lun (Treatise on cold afflictions), written around the end of the second century.

Source: Kiple (ed)

In the west, the relationship with seasons was also noted by Hippocrates and Galen. Hippocrates wrote:

Of winter, pleurisy, pneumonia, coryza, hoarseness, cough, pains of the chest, pains of the ribs and loins, headache, vertigo, and apoplexy.

The Greek physician Pedanius Dioscorides' (1st century AD) work De Materia Medica contains remedies for (possibly) symptoms of colds:

Dioscorides describes radishes as anti-inflammatory for the throat and a cough, and onion juice mixed with honey – still used today – as helpful in discharging mucus. Sulfur was also considered effective: “It helps against cough and internal ulceration if it is eaten with eggs or applied by smoking. … It is useful against catarrh…”

Source: cited in Atzl and Helms

In Egypt, there is the Ebers Papyrus from around 1550 BC:

In addition to 20 different kinds of cough, the cold is also a topic in the Ebers papyrus. The formula for a spell that should drive the cold out of the body reads: “Flow out, fetid nose, flow out, son of fetid nose. Flow out, thou breakest bones, destroyest the skull, and makest ill the seven holes of the head”

Source: Atzl and Helms