There is no such consulship or year. The consulship of the Republic was based on a system of checks on power within a very narrow oligarchical structure, and was rarely truly harmonious. Your question is also incredibly subjective. Any Roman historian could pick a year/consulship to write here, and I could challenge it, and they could challenge any one I pick. There was no hour, minute, and second when the Republic began to "decline." If you want to agree that it declined at all, it was a slow process over several centuries, with roots and causes stretching back even further.

In direct violation of the above, however, I can pick a date. 70 BCE, the first consulship of Gn. Pompeius and M. Crassus. It was not a typical election, of course. Pompey was far too young to stand for or win the consulship, but his election was virtually inevitable. He had commanded consular armies for the previous seven years and had (finally) defeated Sertorius in Spain and won himself a triumph. The war in Spain had been a major problem for the Senate, and its end was a cause for real celebration. In parallel, the uprising of the slaves under Spartacus had been a very close and very disturbing event, and despite some setbacks M. Crassus had ended that as well. Crassus was old enough, rich, and had a recent important victory under his belt. He was the natural choice for the other consul's seat.

Things were (relatively) peaceful for the first time in a long time at the Capitol, and upon the eve of the elections for the consulships of 70, both Pompey and Crassus had armies in Latium awaiting triumphs and ovations without, we think, being particularly menacing (a true change from the past decades). It is unlikely the two men liked each other very much, and Crassus was surely jealous of the younger man's unbelievable successes, but they managed to get along, or at least make it seem so. During their year, they competed with one another in the most harmless way possible: by trying to give the most luxurious celebrations to the Roman people.

In general, the year 70 is marked by a number of initiatives which seem now, and must have seemed then, to be an attempt to return the Roman Republic to some kind of normalcy. The tribunate was rehabilitated from its Sullan cuts, the law courts were reorganized a final time, and Censors were elected after a lapse of 15 years. They set about purging the Senate and taking the census. There were just under a million Romans counted. Pompey, as consul-elect, had promised to address the abuses of Roman governors in the provinces, but nothing came of it. Still, the sentiment was there, and the prosecution of Verres by Cicero in this year served the same purpose. Both consuls declined to go as proconsular governors at the end of their terms.

A year later, in 69, the censors performed the lustrum, a ritual purification of the city. It was a city-wide symbol of renewal which was supposed to be performed regularly, but had lapsed in the chaos of the previous years. In 69 BCE there seemed to be every indication that the strife was over, and that it would be happy and prosperous times ahead for the the Romans. In the west, Sertorius was defeated and his deserters and the followers of Lepidus had been granted general amnestly; in Italy, Spartacus was dead and the slaves were back at their work; in the east, the constant menace of Mithridates was being handled by Lucullus (but not yet complete). None of the Romans in 69 could have known that they were, in fact, undergoing the last Roman lustrum, and that the bloody strife of the 1st century was barely begun.