While Zheng He's voyages were really of short-term consequence, they do indicate that China still possessed significant naval expertise inherited from the navally-oriented Song and Yuan Dynasties. Continued state-sponsored voyages will keep this naval knowledge alive to some degree, but only with a significant expansion in Ming maritime trade would this knowledge really be expanded.
It is possible to see trade as part of a 'prestige' project - the Yuan encouraged maritime trade as part of a strategy to make SE Asia economically dependent on China (and by extension the Khan/Emperor), as well as pave the way for further conquests.
There are several issues with Chinese trade:
1) Balance of payments. Contrary to the common view of Chinese trade with foreigners, the significant volume of Chinese trade during the Song and Yuan period actually caused a balance of payments issue in China, as its gentry classes imported huge quantities of pearls, spices and incense while paying for the goods in cash.
In such a light, the restrictions on trade imposed by the Ming and Qing in later periods was as much about keeping as much cash in the country (largely through creating artificial monopolies, such as limiting the number of trading ports) as it was about foreign influence.
2) Influence of merchants. To sustain Chinese maritime expansion one would also have to nurture a well of naval expertise (sailing, shipbuilding and so on), which can really only be done on a large scale with promotion of maritime trade: indeed, the Chinese merchant class was a critical components of Song and Yuan naval policy, who subsidized and promoted the merchant marine on the condition that they be usable for war.
However, having too strong a merchant class would weaken the state's hold over the country. The Song and Yuan both saw the emergence of a strong 'trade lobby' that regularly lobbied for tariff decreases and state retreat from the economy. In extreme cases you also had merchants becoming powerful local strongmen who could make or break dynasties - Pu Shougeng (an Arab)'s defection to the Yuan after Hangzhou fell basically handed the rest of the Song Dynasty to the Mongols. Even if you don't buy the Neo-Confucian distaste for merchants, this potential threat to state power is also something to consider when thinking about the Chinese restrictions on trade.
3) Priorities. The Song Dynasty was 'forced' into the sea, so to speak, by the closure of the overland caravan trade due to strong nomad confederations on its northern border - the situation became more acute once the navy became the key to the survival of the Southern Song. By the time of the Ming Dynasty China had regained control of its northern frontier and was once again turning towards a continental configuration, where defending the threat of Mongolia and Central Asia on land became the main priority of the state. As such, precious few resources could be spent on the navy, which went into a decline.
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If Zheng He's voyages were followed up on, we'd likely see a greater Chinese interest in the West, especially in India. Certainly the Ming Dynasty would take an interest in establishing important maritime outposts along the India-China trade route such as Palembang (Jiugang), Malacca, and Champa. I think overseas colonization is probably not likely since the Ming already have enough colonizing to do on their south- and north-west frontiers: most likely the Emperors would be satisfied with tribute and submissions of fealty from local leaders.
When the Europeans arrive I think we'd probably see a proxy war over trade-routes, with the Ming and Portugal/Spain intervening regularly to place their leaders on the various SE Asian thrones. Discovery of America... possible, but not likely, since the goods China was interested in were from India and SE Asia, not Europe, so the Chinese don't have much of a reason to go for an Eastern route to Europe.