I think it's first worth looking at the logical people that Alexious could not ask for help. The most immediate sources of military aid were those near the empire with the capacity to raise a large enough army: the Holy Roman Empire, the papacy, the Italians, and the Muslims. All but the papacy were out of the question.

He could not ask the Holy Roman Emperor because the Byzantines refused to accept that the HRE was legitimate. Under Roman law, only the sitting emperor could raise another person to the title of emperor, and no Byzantine emperor had granted that. As far as the Byzantine Empire was concerned, the Holy Roman Emperor was a barbarian usurper. Alexios was more accepting of the HRE than previous Byzantine rulers, but a bad experience with them soured relations anyway. The Byzantines had asked Henry IV for help against the Normans in 1082.. Henry promised to help, took Byzantine money, then spent it all on getting an antipope installed to make him emperor, and didn't help the Byzantines at all. The HRE was out of the question.

He could not ask the Italians, because the Byzantines and the Normans of southern Italy had fallen out over a failed marriage alliance. The Normans had attacked and invaded the western Balkan coast, it is alleged that Alexios and Bohemond fought each other directly at one point in that campaign. In the end, Alexios did get a lot of help from the Normans in the form of Bohemond, but it would have been very awkward to ask for this help directly. Signalling to the Normans that the Byzantine Empire was weak risked an invasion, as the west Balkan coast had long been a target of Norman expansion. Alexios also couldn't ask for help from the powerful city-states because he had nothing to offer them. They already had trade privileges, immunity from taxation within Byzantine territory, and were already rich.

He could not ask the Turks or any Muslim power, since the Turks were the ones attacking him and the rest of the Islamic world was busy attacking itself. Usually, the Muslim powers were happy to provide mercenaries to the empire but relations between Byzantium and the Turks were not exactly at a high point as of the 1090s, and those Islamic powers were a bit busy disintegrating into various bickering fiefdoms.

That's three of the four most immediate sources of help out of the question. That just left the papacy. There was precedent for cooperation between the papacy and the Byzantine Empire on military matters. When the Byzantine Empire was invaded in 1074, the emperor also wrote to the pope to get help. Pope Gregory VII even sent an encyclical letter to be read out in churches across Europe:

We hereby inform you that the bearer of this letter, on his recent return from across the sea [from Palestine], came to Rome to visit us. He repeated what we had heard from many others, that a pagan race had overcome the Christians and with horrible cruelty had devastated everything almost to the walls of Constantinople, and were now governing the conquered lands with tyrannical violence, and that they had slain many thousands of Christians as if they were but sheep. If we love God and wish to be recognized as Christians, we should be filled with grief at the misfortune of this great empire [the Greek] and the murder of so many Christians. But simply to grieve is not our whole duty. The example of our Redeemer and the bond of fraternal love demand that we should lay down our lives to liberate them. "Because he has laid down his life for us: and we ought to lay down our lives for the brethren," [1 John 3:16]. Know, therefore, that we are trusting in the mercy of God and in the power of his might and that we are striving in all possible ways and making preparations to render aid to the Christian empire [the Greek] as quickly as possible. Therefore we beseech you by the faith in which you are united through Christ in the adoption of the sons of God, and by the authority of St. Peter, prince of apostles, we admonish you that you be moved to proper compassion by the wounds and blood of your brethren and the danger of the aforesaid empire and that, for the sake of Christ, you undertake the difficult task of bearing aid to your brethren [the Greeks]. Send messengers to us at once inform us of what God may inspire you to do in this matter.

The almost-crusade of 1074 never got off the ground because relations between the papacy and the Byzantine Empire rapidly deteriorated after a change of emperor, but when Alexios was looking for help it served as an example of how the pope could mobilise military forces. Alexios could also offer the papacy something invaluable: the reunification of the church. Alexios had signalled that he was willing to work to heal the schism of 1054 which split off Constantinople from Rome, which Urban II was eager to do. On top of that, there were stories of pilgrims to Jerusalem being roughed up by Muslims and of churches being looted - the impetus to protect those vulnerable Christians was a powerful one which would have found a very willing reception in the papacy.

The pope was therefore the most logical person to ask for help. The Pope had the influence to put together an army, the motivation to help eastern Christians and bring Constantinople back under the control of the papacy, and in any case was the only realistic option to get an army together.

Alexios also attempted to get the aid of many secular leaders, most notably Robert I of Flanders. In the 1080s, Robert had supplied 500 knights, and in 1091 Alexios had asked him to put together a larger military expedition to help the empire against the Turks. We are told by chroniclers that Alexios had sent out letters to a lot of people, not just the papacy, and although none of them survive we have no reason to doubt that he would have done this. However, these letters got knights to drip into the empire when Alexios actually needed an army. Only the papacy could really galvanise the disparate peoples of Europe into an army.