The short term effects of the Space Race greatly increased tensions between the US and USSR through new missile technologies and surveillance techniques, but the long term effects were actually beneficial. It sparked an increase on education spending and improved the maths and sciences. Also the technologies developed during the Space Race ended up being used for other purposes outside of military use.
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Even before the space race began there were negotiations between the US and USSR about using space for peaceful purposes. They signed a document saying there will be no WMD’s in space, no military presence in space, and no government can claim a celestial body. This actually ended up exacerbating the cold war from the start. This gave both nations more of an incentive to expand their influence on earth now that space was out of the question.