There is no right or wrong way to place a greeting card inside an envelope. Whether it is placed with the crease side up or down is really just a matter of preference.
Although this topic is debated among etiquette experts, there is no universal answer to card placement in envelopes. However, there are some practical reasons for inserting cards in a particular manner.
Many people open cards from the back side, whether they rip the top open or use a letter opener. If a card is in the envelope with the front of it facing backwards, the front of the card, the side with the images, is immediately visible.
Another point to consider is whether or not the card contains money, a check or a gift card. If a card is inserted crease side up in an envelope, it is more likely anything inside the card will fall out when the recipient removes it from the envelope. This could cause the contents to be lost or difficult to find.
For these reasons, although there is no formal mandate, it is considered best practice to place a card crease side down in an envelope, with the front of the card facing the back of the envelope.