The in-game Japanese text of Yoshi's
Trophy in
Super Smash Bros. Melee states that Yoshis reproduce asexually, meaning that they reproduce without a mate and are neither male nor female. This is supported by the
Chef minigame in
Game & Watch Galleries 2 and
4; after being fed long enough, the Yoshi produces an egg that eventually hatches into another Yoshi. The baby then takes its parent's place and proceeds to eat enough food to turn into an adult, eventually producing a fertile egg of its own, which then continues the cycle.
Other sources have also contradicted the asexual depiction of Yoshis. At least one of the Yoshis in
Paper Mario refers to his "son", and in
Paper Mario: The Thousand-Year Door, the Yoshi who sometimes appears on-board the
Excess Express expresses his hatred of business trips due to them keeping him away from his "lovely wife". The original
Mario Party also features a pair of Yoshis which appear to be one male and one female, although the exact nature of their relationship was not revealed. Additionally, while Japanese language rarely involves gender-specific pronouns, the character Yoshi is consistently referred to with masculine pronouns in translation, yet laying eggs is one of his trademarks. This apparent contradiction was addressed in
Super Smash Bros. Brawl, where
Snake assumes that Yoshi is a female because of his egg-laying abilities, only to be corrected by
Otacon (both of whom use "it" rather than "he" or "she" when referencing Yoshi). It is addressed again in
Super Smash Bros. for Wii U, where
Viridi remarks that there is nothing natural about a male who lays eggs while
Palutena asserts that all living beings contain both male and female elements.
Tl;Dr: Nintendo can't make up their minds. They're sometimes genderless, sometimes they have genders. That said, they reproduce assexualy.