Very tentatively.

Obviously, the Arab states viewed it unfavorably, since they invaded the day it took effect (a few hours after it was issued, they declared their invasion). The major powers handled it a little differently.

President Truman, for his part, decided almost immediately to grant Israel de facto (concerning fact, roughly) recognition. It took him about 11 minutes, if memory serves. Truman said this:

This government has been informed that a Jewish state has been proclaimed in Palestine, and recognition has been requested by the provisional government thereof. The United States recognizes the provisional government as the de facto authority of the new State of Israel.

The Soviet Union granted de jure (concerning law) recognition 3 days after the declaration. This type of recognition, considered stronger, came in a letter saying this:

Confirming receipt of your telegram of May 16, in which you inform the Government of the USSR of the proclamation, on the basis of the resolution of the United Nations Assembly of November 29, 1947, of the creation in Palestine of the independent State of Israel and make request for the recognition of the State of Israel and its provisional government by the USSR, I inform you in this letter that the Government of the USSR has decided to recognize officially the State of Israel and its provisional government.

In context, you'd think the two superpowers recognizing Israel would be enough to get most of their allies to follow suit. In a way, it did encourage others. At least 50 countries recognized Israel by the end of the end of the war (well, the Syrian ceasefire anyways, which came last) on July 20. The West, it seemed, took a little longer to actually recognize Israel (outside of the United States). The "great powers" of the West (France, UK, United States) did not provide recognition (outside of the US's lone de facto recognition) until at January of 1949 for France, and May 1949 for the UK.

It's rather hard to judge general reaction to the Declaration, if only because the United Nations did not pass a resolution on it immediately, or otherwise react. Israel was admitted to the United Nations on May 11, 1949 (while still at war with only Syria, really), in a vote that came out with 37 in favor, 9 abstaining, and 12 against. The vote went like this:

Yes:

Y ARGENTINA
Y AUSTRALIA
Y BOLIVIA
Y BYELORUSSIAN SSR
Y CANADA
Y CHILE
Y CHINA
Y COLOMBIA
Y COSTA RICA
Y CUBA
Y CZECHOSLOVAKIA
Y DOMINICAN REPUBLIC
Y ECUADOR
Y FRANCE
Y GUATEMALA
Y HAITI
Y HONDURAS
Y ICELAND
Y LIBERIA
Y LUXEMBOURG
Y MEXICO
Y NETHERLANDS
Y NEW ZEALAND
Y NICARAGUA
Y NORWAY
Y PANAMA
Y PARAGUAY
Y PERU
Y PHILIPPINES
Y POLAND
Y UKRAINIAN SSR
Y UNION OF SOUTH AFRICA
Y USSR
Y UNITED STATES
Y URUGUAY
Y VENEZUELA
Y YUGOSLAVIA

No:

N AFGHANISTAN
N BURMA
N EGYPT
N ETHIOPIA
N INDIA
N IRAN
N IRAQ
N LEBANON
N PAKISTAN
N SAUDI ARABIA
N SYRIA
N YEMEN

Abstaining:

A BELGIUM
A BRAZIL
A DENMARK
A EL SALVADOR
A GREECE
A SIAM
A SWEDEN
A TURKEY
A UNITED KINGDOM

Interestingly enough, this vote is not the same as the vote for the UN Partition Plan that marked the start of the Civil War in the area (that preceded the declaration). It gives you an idea of how lukewarm support was even a year after the declaration, and how little it had changed. The changes were:

China - No to Yes.
Cuba - No to Yes.
Greece - No to Abstain.
Turkey - No to Abstain.
Argentina - Abstain to Yes.
Chile - Abstain to Yes.
Colombia - Abstain to Yes.
Honduras - Abstain to Yes.
Mexico - Abstain to Yes.
Yugoslavia - Abstain to Yes.
Ethiopia - Abstain to No.
Brazil - Yes to Abstain.
Belgium - Yes to Abstain.
Denmark - Yes to Abstain.
Sweden - Yes to Abstain.

15 votes changed. Of them, 10 were "positive" shifts (ie. No to Abstain, or No to Yes, or Abstain to Yes), while 5 were "negative shifts" (ie. Abstain to No, Yes to No, Yes to Abstain).

For 15 nations to change their votes of the 57 members at the time of the Partition Plan was pretty indicative of the different feelings going around, but it also shows the mild and fickle nature of that support.

Unfortunately I've been unable to find more on how specific nations reacted outside of what I've presented here, at least to the declaration itself. The recognition timeline should give you an idea of what the community thought, though, and the UN vote to admit Israel should as well.